[Scipy-svn] r4766 - trunk/scipy/stats
scipy-svn at scipy.org
scipy-svn at scipy.org
Fri Oct 3 14:57:21 EDT 2008
Author: oliphant
Date: 2008-10-03 13:57:20 -0500 (Fri, 03 Oct 2008)
New Revision: 4766
Modified:
trunk/scipy/stats/continuous.lyx
trunk/scipy/stats/distributions.py
Log:
Improve docstring of lognorm a bit.
Modified: trunk/scipy/stats/continuous.lyx
===================================================================
--- trunk/scipy/stats/continuous.lyx 2008-10-03 14:31:41 UTC (rev 4765)
+++ trunk/scipy/stats/continuous.lyx 2008-10-03 18:57:20 UTC (rev 4766)
@@ -1,18 +1,19 @@
-#LyX 1.3 created this file. For more info see http://www.lyx.org/
-\lyxformat 221
+#LyX 1.4.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 245
+\begin_document
+\begin_header
\textclass article
\language english
\inputencoding auto
\fontscheme default
\graphics default
\paperfontsize default
-\spacing single
-\papersize Default
-\paperpackage a4
-\use_geometry 1
-\use_amsmath 1
-\use_natbib 0
-\use_numerical_citations 0
+\spacing single
+\papersize default
+\use_geometry true
+\use_amsmath 2
+\cite_engine basic
+\use_bibtopic false
\paperorientation portrait
\leftmargin 1in
\topmargin 1in
@@ -23,43 +24,49 @@
\paragraph_separation indent
\defskip medskip
\quotes_language english
-\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle default
+\tracking_changes false
+\output_changes true
+\end_header
-\layout Title
+\begin_body
+\begin_layout Title
Continuous Statistical Distributions
-\layout Section
+\end_layout
+\begin_layout Section
Overview
-\layout Standard
+\end_layout
+\begin_layout Standard
All distributions will have location (L) and Scale (S) parameters along
with any shape parameters needed, the names for the shape parameters will
vary.
Standard form for the distributions will be given where
\begin_inset Formula $L=0.0$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=1.0.$
-\end_inset
+\end_inset
The nonstandard forms can be obtained for the various functions using (note
\begin_inset Formula $U$
-\end_inset
+\end_inset
is a standard uniform random variate).
-\layout Standard
-\align center
+\end_layout
-\size small
+\begin_layout Standard
+\align center
-\begin_inset Tabular
+\size small
+\begin_inset Tabular
<lyxtabular version="3" rows="16" columns="3">
<features>
<column alignment="center" valignment="top" leftline="true" width="0pt">
@@ -69,591 +76,612 @@
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Function Name
+\end_layout
-Function Name
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Standard Function
+\end_layout
-Standard Function
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Transformation
+\end_layout
-Transformation
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Cumulative Distribution Function (CDF)
+\end_layout
-Cumulative Distribution Function (CDF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $F\left(x\right)$
+\end_inset
-\begin_inset Formula $F\left(x\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $F\left(x;L,S\right)=F\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
-\begin_inset Formula $F\left(x;L,S\right)=F\left(\frac{\left(x-L\right)}{S}\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Probability Density Function (PDF)
+\end_layout
-Probability Density Function (PDF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $f\left(x\right)=F^{\prime}\left(x\right)$
+\end_inset
-\begin_inset Formula $f\left(x\right)=F^{\prime}\left(x\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $f\left(x;L,S\right)=\frac{1}{S}f\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
-\begin_inset Formula $f\left(x;L,S\right)=\frac{1}{S}f\left(\frac{\left(x-L\right)}{S}\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Percent Point Function (PPF)
+\end_layout
-Percent Point Function (PPF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $G\left(q\right)=F^{-1}\left(q\right)$
+\end_inset
-\begin_inset Formula $G\left(q\right)=F^{-1}\left(q\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $G\left(q;L,S\right)=L+SG\left(q\right)$
+\end_inset
-\begin_inset Formula $G\left(q;L,S\right)=L+SG\left(q\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Probability Sparsity Function (PSF)
+\end_layout
-Probability Sparsity Function (PSF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
-
-
+\begin_layout Standard
\begin_inset Formula $g\left(q\right)=G^{\prime}\left(q\right)$
-\end_inset
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $g\left(q;L,S\right)=Sg\left(q\right)$
+\end_inset
-\begin_inset Formula $g\left(q;L,S\right)=Sg\left(q\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Hazard Function (HF)
+\end_layout
-Hazard Function (HF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $h_{a}\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}$
+\end_inset
-\begin_inset Formula $h_{a}\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $h_{a}\left(x;L,S\right)=\frac{1}{S}h_{a}\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
-\begin_inset Formula $h_{a}\left(x;L,S\right)=\frac{1}{S}h_{a}\left(\frac{\left(x-L\right)}{S}\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Cumulative Hazard Functon (CHF)
+\end_layout
-Cumulative Hazard Functon (CHF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
-
-
+\begin_layout Standard
\begin_inset Formula $H_{a}\left(x\right)=$
-\end_inset
+\end_inset
\begin_inset Formula $\log\frac{1}{1-F\left(x\right)}$
-\end_inset
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $H_{a}\left(x;L,S\right)=H_{a}\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
-\begin_inset Formula $H_{a}\left(x;L,S\right)=H_{a}\left(\frac{\left(x-L\right)}{S}\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Survival Function (SF)
+\end_layout
-Survival Function (SF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $S\left(x\right)=1-F\left(x\right)$
+\end_inset
-\begin_inset Formula $S\left(x\right)=1-F\left(x\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $S\left(x;L,S\right)=S\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
-\begin_inset Formula $S\left(x;L,S\right)=S\left(\frac{\left(x-L\right)}{S}\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Inverse Survival Function (ISF)
+\end_layout
-Inverse Survival Function (ISF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)$
+\end_inset
-\begin_inset Formula $Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $Z\left(\alpha;L,S\right)=L+SZ\left(\alpha\right)$
+\end_inset
-\begin_inset Formula $Z\left(\alpha;L,S\right)=L+SZ\left(\alpha\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Moment Generating Function (MGF)
+\end_layout
-Moment Generating Function (MGF)
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $M_{Y}\left(t\right)=E\left[e^{Yt}\right]$
+\end_inset
-\begin_inset Formula $M_{Y}\left(t\right)=E\left[e^{Yt}\right]$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $M_{X}\left(t\right)=e^{Lt}M_{Y}\left(St\right)$
+\end_inset
-\begin_inset Formula $M_{X}\left(t\right)=e^{Lt}M_{Y}\left(St\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Random Variates
+\end_layout
-Random Variates
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $Y=G\left(U\right)$
+\end_inset
-\begin_inset Formula $Y=G\left(U\right)$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $X=L+SY$
+\end_inset
-\begin_inset Formula $X=L+SY$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+(Differential) Entropy
+\end_layout
-(Differential) Entropy
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $h\left[Y\right]=-\int f\left(y\right)\log f\left(y\right)dy$
+\end_inset
-\begin_inset Formula $h\left[Y\right]=-\int f\left(y\right)\log f\left(y\right)dy$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $h\left[X\right]=h\left[Y\right]+\log S$
+\end_inset
-\begin_inset Formula $h\left[X\right]=h\left[Y\right]+\log S$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+(Non-central) Moments
+\end_layout
-(Non-central) Moments
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $\mu_{n}^{\prime}=E\left[Y^{n}\right]$
+\end_inset
-\begin_inset Formula $\mu_{n}^{\prime}=E\left[Y^{n}\right]$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
-
-
+\begin_layout Standard
\begin_inset Formula $E\left[X^{n}\right]=L^{n}\sum_{k=0}^{N}\left(\begin{array}{c}
n\\
k\end{array}\right)\left(\frac{S}{L}\right)^{k}\mu_{k}^{\prime}$
-\end_inset
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+Central Moments
+\end_layout
-Central Moments
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $\mu_{n}=E\left[\left(Y-\mu\right)^{n}\right]$
+\end_inset
-\begin_inset Formula $\mu_{n}=E\left[\left(Y-\mu\right)^{n}\right]$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $E\left[\left(X-\mu_{X}\right)^{n}\right]=S^{n}\mu_{n}$
+\end_inset
-\begin_inset Formula $E\left[\left(X-\mu_{X}\right)^{n}\right]=S^{n}\mu_{n}$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+mean (mode, median), var
+\end_layout
-mean (mode, median), var
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $\mu,\,\mu_{2}$
+\end_inset
-\begin_inset Formula $\mu,\,\mu_{2}$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
-
-
+\begin_layout Standard
\begin_inset Formula $L+S\mu,\, S^{2}\mu_{2}$
-\end_inset
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
</row>
<row topline="true" bottomline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+skewness, kurtosis
+\end_layout
-skewness, kurtosis
-\end_inset
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
-
-
+\begin_layout Standard
\begin_inset Formula $\gamma_{1}=\frac{\mu_{3}}{\left(\mu_{2}\right)^{3/2}},\,$
-\end_inset
+\end_inset
\begin_inset Formula $\gamma_{2}=\frac{\mu_{4}}{\left(\mu_{2}\right)^{2}}-3$
-\end_inset
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $\gamma_{1},\,\gamma_{2}$
+\end_inset
-\begin_inset Formula $\gamma_{1},\,\gamma_{2}$
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
</lyxtabular>
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-\SpecialChar ~
+\begin_layout Standard
+\InsetSpace ~
-\layout Subsection
+\end_layout
+\begin_layout Subsection
Moments
-\layout Standard
+\end_layout
+\begin_layout Standard
Non-central moments are defined using the PDF
\begin_inset Formula \[
\mu_{n}^{\prime}=\int_{-\infty}^{\infty}x^{n}f\left(x\right)dx.\]
-\end_inset
+\end_inset
Note, that these can always be computed using the PPF.
Substitute
\begin_inset Formula $x=G\left(q\right)$
-\end_inset
+\end_inset
in the above equation and get
\begin_inset Formula \[
\mu_{n}^{\prime}=\int_{0}^{1}G^{n}\left(q\right)dq\]
-\end_inset
+\end_inset
which may be easier to compute numerically.
Note that
\begin_inset Formula $q=F\left(x\right)$
-\end_inset
+\end_inset
so that
\begin_inset Formula $dq=f\left(x\right)dx.$
-\end_inset
+\end_inset
Central moments are computed similarly
\begin_inset Formula $\mu=\mu_{1}^{\prime}$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -663,7 +691,7 @@
n\\
k\end{array}\right)\left(-\mu\right)^{k}\mu_{n-k}^{\prime}\end{eqnarray*}
-\end_inset
+\end_inset
In particular
\begin_inset Formula \begin{eqnarray*}
@@ -672,157 +700,162 @@
\mu_{4} & = & \mu_{4}^{\prime}-4\mu\mu_{3}^{\prime}+6\mu^{2}\mu_{2}^{\prime}-3\mu^{4}\\
& = & \mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}\end{eqnarray*}
-\end_inset
+\end_inset
Skewness is defined as
\begin_inset Formula \[
\gamma_{1}=\sqrt{\beta_{1}}=\frac{\mu_{3}}{\mu_{2}^{3/2}}\]
-\end_inset
+\end_inset
while (Fisher) kurtosis is
\begin_inset Formula \[
\gamma_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3,\]
-\end_inset
+\end_inset
so that a normal distribution has a kurtosis of zero.
-\layout Subsection
+\end_layout
+\begin_layout Subsection
Median and mode
-\layout Standard
+\end_layout
+\begin_layout Standard
The median,
\begin_inset Formula $m_{n}$
-\end_inset
+\end_inset
is defined as the point at which half of the density is on one side and
half on the other.
In other words,
\begin_inset Formula $F\left(m_{n}\right)=\frac{1}{2}$
-\end_inset
+\end_inset
so that
\begin_inset Formula \[
m_{n}=G\left(\frac{1}{2}\right).\]
-\end_inset
+\end_inset
In addition, the mode,
\begin_inset Formula $m_{d}$
-\end_inset
+\end_inset
, is defined as the value for which the probability density function reaches
it's peak
\begin_inset Formula \[
m_{d}=\arg\max_{x}f\left(x\right).\]
-\end_inset
+\end_inset
-\layout Subsection
+\end_layout
+\begin_layout Subsection
Fitting data
-\layout Standard
+\end_layout
+\begin_layout Standard
To fit data to a distribution, maximizing the likelihood function is common.
Alternatively, some distributions have well-known minimum variance unbiased
estimators.
These will be chosen by default, but the likelihood function will always
be available for minimizing.
-\layout Standard
+\end_layout
+\begin_layout Standard
If
\begin_inset Formula $f\left(x;\boldsymbol{\theta}\right)$
-\end_inset
+\end_inset
is the PDF of a random-variable where
\begin_inset Formula $\boldsymbol{\theta}$
-\end_inset
+\end_inset
is a vector of parameters (
-\emph on
+\emph on
e.g.
\begin_inset Formula $L$
-\end_inset
+\end_inset
-\emph default
+\emph default
and
\begin_inset Formula $S$
-\end_inset
+\end_inset
), then for a collection of
\begin_inset Formula $N$
-\end_inset
+\end_inset
independent samples from this distribution, the joint distribution the
random vector
\begin_inset Formula $\mathbf{x}$
-\end_inset
+\end_inset
is
\begin_inset Formula \[
f\left(\mathbf{x};\boldsymbol{\theta}\right)=\prod_{i=1}^{N}f\left(x_{i};\boldsymbol{\theta}\right).\]
-\end_inset
+\end_inset
The maximum likelihood estimate of the parameters
\begin_inset Formula $\boldsymbol{\theta}$
-\end_inset
+\end_inset
are the parameters which maximize this function with
\begin_inset Formula $\mathbf{x}$
-\end_inset
+\end_inset
fixed and given by the data:
\begin_inset Formula \begin{eqnarray*}
\boldsymbol{\theta}_{es} & = & \arg\max_{\boldsymbol{\theta}}f\left(\mathbf{x};\boldsymbol{\theta}\right)\\
& = & \arg\min_{\boldsymbol{\theta}}l_{\mathbf{x}}\left(\boldsymbol{\theta}\right).\end{eqnarray*}
-\end_inset
+\end_inset
Where
\begin_inset Formula \begin{eqnarray*}
l_{\mathbf{x}}\left(\boldsymbol{\theta}\right) & = & -\sum_{i=1}^{N}\log f\left(x_{i};\boldsymbol{\theta}\right)\\
& = & -N\overline{\log f\left(x_{i};\boldsymbol{\theta}\right)}\end{eqnarray*}
-\end_inset
+\end_inset
Note that if
\begin_inset Formula $\boldsymbol{\theta}$
-\end_inset
+\end_inset
includes only shape parameters, the location and scale-parameters can be
fit by replacing
\begin_inset Formula $x_{i}$
-\end_inset
+\end_inset
with
\begin_inset Formula $\left(x_{i}-L\right)/S$
-\end_inset
+\end_inset
in the log-likelihood function adding
\begin_inset Formula $N\log S$
-\end_inset
+\end_inset
and minimizing, thus
\begin_inset Formula \begin{eqnarray*}
l_{\mathbf{x}}\left(L,S;\boldsymbol{\theta}\right) & = & N\log S-\sum_{i=1}^{N}\log f\left(\frac{x_{i}-L}{S};\boldsymbol{\theta}\right)\\
& = & N\log S+l_{\frac{\mathbf{x}-S}{L}}\left(\boldsymbol{\theta}\right)\end{eqnarray*}
-\end_inset
+\end_inset
If desired, sample estimates for
\begin_inset Formula $L$
-\end_inset
+\end_inset
and
\begin_inset Formula $S$
-\end_inset
+\end_inset
(not necessarily maximum likelihood estimates) can be obtained from samples
estimates of the mean and variance using
@@ -830,185 +863,194 @@
\hat{S} & = & \sqrt{\frac{\hat{\mu}_{2}}{\mu_{2}}}\\
\hat{L} & = & \hat{\mu}-\hat{S}\mu\end{eqnarray*}
-\end_inset
+\end_inset
where
\begin_inset Formula $\mu$
-\end_inset
+\end_inset
and
\begin_inset Formula $\mu_{2}$
-\end_inset
+\end_inset
are assumed known as the mean and variance of the
-\series bold
+\series bold
untransformed
-\series default
+\series default
distribution (when
\begin_inset Formula $L=0$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=1$
-\end_inset
+\end_inset
) and
\begin_inset Formula \begin{eqnarray*}
\hat{\mu} & = & \frac{1}{N}\sum_{i=1}^{N}x_{i}=\bar{\mathbf{x}}\\
\hat{\mu}_{2} & = & \frac{1}{N-1}\sum_{i=1}^{N}\left(x_{i}-\hat{\mu}\right)^{2}=\frac{N}{N-1}\overline{\left(\mathbf{x}-\bar{\mathbf{x}}\right)^{2}}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Subsection
+\end_layout
+\begin_layout Subsection
Standard notation for mean
-\layout Standard
+\end_layout
+\begin_layout Standard
We will use
\begin_inset Formula \[
\overline{y\left(\mathbf{x}\right)}=\frac{1}{N}\sum_{i=1}^{N}y\left(x_{i}\right)\]
-\end_inset
+\end_inset
where
\begin_inset Formula $N$
-\end_inset
+\end_inset
should be clear from context as the number of samples
\begin_inset Formula $x_{i}$
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Alpha
-\layout Standard
+\end_layout
+\begin_layout Standard
One shape parameters
\begin_inset Formula $\alpha>0$
-\end_inset
+\end_inset
(paramter
\begin_inset Formula $\beta$
-\end_inset
+\end_inset
in DATAPLOT is a scale-parameter).
Standard form is
\begin_inset Formula $x>0:$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha\right) & = & \frac{1}{x^{2}\Phi\left(\alpha\right)\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\alpha-\frac{1}{x}\right)^{2}\right)\\
F\left(x;\alpha\right) & = & \frac{\Phi\left(\alpha-\frac{1}{x}\right)}{\Phi\left(\alpha\right)}\\
G\left(q;\alpha\right) & = & \left[\alpha-\Phi^{-1}\left(q\Phi\left(\alpha\right)\right)\right]^{-1}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\frac{1}{\Phi\left(a\right)\sqrt{2\pi}}\int_{0}^{\infty}\frac{e^{xt}}{x^{2}}\exp\left(-\frac{1}{2}\left(\alpha-\frac{1}{x}\right)^{2}\right)dx\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
+\begin_layout Standard
No moments?
\begin_inset Formula \[
l_{\mathbf{x}}\left(\alpha\right)=N\log\left[\Phi\left(\alpha\right)\sqrt{2\pi}\right]+2N\overline{\log\mathbf{x}}+\frac{N}{2}\alpha^{2}-\alpha\overline{\mathbf{x}^{-1}}+\frac{1}{2}\overline{\mathbf{x}^{-2}}\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Anglit
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined over
\begin_inset Formula $x\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \sin\left(2x+\frac{\pi}{2}\right)=\cos\left(2x\right)\\
F\left(x\right) & = & \sin^{2}\left(x+\frac{\pi}{4}\right)\\
G\left(q\right) & = & \arcsin\left(\sqrt{q}\right)-\frac{\pi}{4}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & 0\\
\mu_{2} & = & \frac{\pi^{2}}{16}-\frac{1}{2}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & -2\frac{\pi^{4}-96}{\left(\pi^{2}-8\right)^{2}}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & 1-\log2\\
& \approx & 0.30685281944005469058\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
M\left(t\right) & = & \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\cos\left(2x\right)e^{xt}dx\\
& = & \frac{4\cosh\left(\frac{\pi t}{4}\right)}{t^{2}+4}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
l_{\mathbf{x}}\left(\cdot\right)=-N\overline{\log\left[\cos\left(2\mathbf{x}\right)\right]}\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Arcsine
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined over
\begin_inset Formula $x\in\left(0,1\right)$
-\end_inset
+\end_inset
.
To get the JKB definition put
\begin_inset Formula $x=\frac{u+1}{2}.$
-\end_inset
+\end_inset
i.e.
\begin_inset Formula $L=-1$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=2.$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1016,20 +1058,20 @@
F\left(x\right) & = & \frac{2}{\pi}\arcsin\left(\sqrt{x}\right)\\
G\left(q\right) & = & \sin^{2}\left(\frac{\pi}{2}q\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=E^{t/2}I_{0}\left(\frac{t}{2}\right)\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
\mu_{n}^{\prime} & = & \frac{1}{\pi}\int_{0}^{1}dx\, x^{n-1/2}\left(1-x\right)^{-1/2}\\
& = & \frac{1}{\pi}B\left(\frac{1}{2},n+\frac{1}{2}\right)=\frac{\left(2n-1\right)!!}{2^{n}n!}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1038,45 +1080,47 @@
\gamma_{1} & = & 0\\
\gamma_{2} & = & -\frac{3}{2}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]\approx-0.24156447527049044468\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
l_{\mathbf{x}}\left(\cdot\right)=N\log\pi+\frac{N}{2}\overline{\log\mathbf{x}}+\frac{N}{2}\overline{\log\left(1-\mathbf{x}\right)}\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Beta
-\layout Standard
+\end_layout
+\begin_layout Standard
Two shape parameters
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
a,b>0\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,b\right) & = & \frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}x^{a-1}\left(1-x\right)^{b-1}I_{\left(0,1\right)}\left(x\right)\\
F\left(x;a,b\right) & = & \int_{0}^{x}f\left(y;a,b\right)dy=I\left(x,a,b\right)\\
@@ -1088,66 +1132,68 @@
\gamma_{2} & = & \frac{6\left(a^{3}+a^{2}\left(1-2b\right)+b^{2}\left(b+1\right)-2ab\left(b+2\right)\right)}{ab\left(a+b+2\right)\left(a+b+3\right)}\\
m_{d} & = & \frac{\left(a-1\right)}{\left(a+b-2\right)}\, a+b\neq2\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula $f\left(x;a,1\right)$
-\end_inset
+\end_inset
is also called the Power-function distribution.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
l_{\mathbf{x}}\left(a,b\right)=-N\log\Gamma\left(a+b\right)+N\log\Gamma\left(a\right)+N\log\Gamma\left(b\right)-N\left(a-1\right)\overline{\log\mathbf{x}}-N\left(b-1\right)\overline{\log\left(1-\mathbf{x}\right)}\]
-\end_inset
+\end_inset
All of the
\begin_inset Formula $x_{i}\in\left[0,1\right]$
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Beta Prime
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined over
\begin_inset Formula $0<x<\infty.$
-\end_inset
+\end_inset
\begin_inset Formula $\alpha,\beta>0.$
-\end_inset
+\end_inset
(Note the CDF evaluation uses Eq.
3.194.1 on pg.
313 of Gradshteyn & Ryzhik (sixth edition).
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha,\beta\right) & = & \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha-1}\left(1+x\right)^{-\alpha-\beta}\\
F\left(x;\alpha,\beta\right) & = & \frac{\Gamma\left(\alpha+\beta\right)}{\alpha\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha}\,_{2}F_{1}\left(\alpha+\beta,\alpha;1+\alpha;-x\right)\\
G\left(q;\alpha,\beta\right) & = & F^{-1}\left(x;\alpha,\beta\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\left\{ \begin{array}{ccc}
\frac{\Gamma\left(n+\alpha\right)\Gamma\left(\beta-n\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}=\frac{\left(\alpha\right)_{n}}{\left(\beta-n\right)_{n}} & & \beta>n\\
\infty & & \textrm{otherwise}\end{array}\right.\]
-\end_inset
+\end_inset
Therefore,
\begin_inset Formula \begin{eqnarray*}
@@ -1157,20 +1203,21 @@
\gamma_{2} & = & \frac{\mu_{4}}{\mu_{2}^{2}}-3\\
\mu_{4} & = & \frac{\alpha\left(\alpha+1\right)\left(\alpha+2\right)\left(\alpha+3\right)}{\left(\beta-4\right)\left(\beta-3\right)\left(\beta-2\right)\left(\beta-1\right)}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}\quad\beta>4\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Bradford
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
c & > & 0\\
k & = & \log\left(1+c\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1185,37 +1232,38 @@
m_{d} & = & 0\\
m_{n} & = & \sqrt{1+c}-1\end{eqnarray*}
-\end_inset
+\end_inset
where
\begin_inset Formula $\textrm{Ei}\left(\textrm{z}\right)$
-\end_inset
+\end_inset
is the exponential integral function.
Also
\begin_inset Formula \[
h\left[X\right]=\frac{1}{2}\log\left(1+c\right)-\log\left(\frac{c}{\log\left(1+c\right)}\right)\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Burr
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
c & > & 0\\
d & > & 0\\
k & = & \Gamma\left(d\right)\Gamma\left(1-\frac{2}{c}\right)\Gamma\left(\frac{2}{c}+d\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+d\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c,d\right) & = & \frac{cd}{x^{c+1}\left(1+x^{-c}\right)^{d+1}}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x;c,d\right) & = & \left(1+x^{-c}\right)^{-d}\\
@@ -1230,15 +1278,16 @@
m_{d} & = & \left(\frac{cd-1}{c+1}\right)^{1/c}\,\textrm{if }cd>1\,\textrm{otherwise }0\\
m_{n} & = & \left(2^{1/d}-1\right)^{-1/c}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Cauchy
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{\pi\left(1+x^{2}\right)}\\
F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\tan^{-1}x\\
@@ -1246,7 +1295,7 @@
m_{d} & = & 0\\
m_{n} & = & 0\end{eqnarray*}
-\end_inset
+\end_inset
No finite moments.
This is the t distribution with one degree of freedom.
@@ -1255,35 +1304,37 @@
h\left[X\right] & = & \log\left(4\pi\right)\\
& \approx & 2.5310242469692907930.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Chi
-\layout Standard
+\end_layout
+\begin_layout Standard
Generated by taking the (positive) square-root of chi-squared variates.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\nu\right) & = & \frac{x^{\nu-1}e^{-x^{2}/2}}{2^{\nu/2-1}\Gamma\left(\frac{\nu}{2}\right)}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x;\nu\right) & = & \Gamma\left(\frac{\nu}{2},\frac{x^{2}}{2}\right)\\
G\left(\alpha;\nu\right) & = & \sqrt{2\Gamma^{-1}\left(\frac{\nu}{2},\alpha\right)}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\Gamma\left(\frac{v}{2}\right)\,_{1}F_{1}\left(\frac{v}{2};\frac{1}{2};\frac{t^{2}}{2}\right)+\frac{t}{\sqrt{2}}\Gamma\left(\frac{1+\nu}{2}\right)\,_{1}F_{1}\left(\frac{1+\nu}{2};\frac{3}{2};\frac{t^{2}}{2}\right)\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{\sqrt{2}\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)}\\
\mu_{2} & = & \nu-\mu^{2}\\
@@ -1292,50 +1343,53 @@
m_{d} & = & \sqrt{\nu-1}\quad\nu\geq1\\
m_{n} & = & \sqrt{2\Gamma^{-1}\left(\frac{\nu}{2},\frac{1}{2}\right)}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Chi-squared
-\layout Standard
+\end_layout
+\begin_layout Standard
This is the gamma distribution with
\begin_inset Formula $L=0.0$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=2.0$
-\end_inset
+\end_inset
and
\begin_inset Formula $\alpha=\nu/2$
-\end_inset
+\end_inset
where
\begin_inset Formula $\nu$
-\end_inset
+\end_inset
is called the degrees of freedom.
If
\begin_inset Formula $Z_{1}\ldots Z_{\nu}$
-\end_inset
+\end_inset
are all standard normal distributions, then
\begin_inset Formula $W=\sum_{k}Z_{k}^{2}$
-\end_inset
+\end_inset
has (standard) chi-square distribution with
\begin_inset Formula $\nu$
-\end_inset
+\end_inset
degrees of freedom.
-\layout Standard
+\end_layout
+\begin_layout Standard
The standard form (most often used in standard form only) is
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1343,13 +1397,13 @@
F\left(x;\alpha\right) & = & \Gamma\left(\frac{\nu}{2},\frac{x}{2}\right)\\
G\left(q;\alpha\right) & = & 2\Gamma^{-1}\left(\frac{\nu}{2},q\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\frac{\Gamma\left(\frac{\nu}{2}\right)}{\left(\frac{1}{2}-t\right)^{\nu/2}}\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1359,19 +1413,21 @@
\gamma_{2} & = & \frac{12}{\nu}\\
m_{d} & = & \frac{\nu}{2}-1\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Cosine
-\layout Standard
+\end_layout
+\begin_layout Standard
Approximation to the normal distribution.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{2\pi}\left[1+\cos x\right]I_{\left[-\pi,\pi\right]}\left(x\right)\\
F\left(x\right) & = & \frac{1}{2\pi}\left[\pi+x+\sin x\right]I_{\left[-\pi,\pi\right]}\left(x\right)+I_{\left(\pi,\infty\right)}\left(x\right)\\
@@ -1382,33 +1438,35 @@
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{-6\left(\pi^{4}-90\right)}{5\left(\pi^{2}-6\right)^{2}}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(4\pi\right)-1\\
& \approx & 1.5310242469692907930.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Double Gamma
-\layout Standard
+\end_layout
+\begin_layout Standard
The double gamma is the signed version of the Gamma distribution.
For
\begin_inset Formula $\alpha>0:$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha\right) & = & \frac{1}{2\Gamma\left(\alpha\right)}\left|x\right|^{\alpha-1}e^{-\left|x\right|}\\
F\left(x;\alpha\right) & = & \left\{ \begin{array}{ccc}
@@ -1418,21 +1476,21 @@
-\Gamma^{-1}\left(\alpha,\left|2q-1\right|\right) & & q\leq\frac{1}{2}\\
\Gamma^{-1}\left(\alpha,\left|2q-1\right|\right) & & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\frac{1}{2\left(1-t\right)^{a}}+\frac{1}{2\left(1+t\right)^{a}}\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu=m_{n} & = & 0\\
\mu_{2} & = & \alpha\left(\alpha+1\right)\\
@@ -1440,24 +1498,28 @@
\gamma_{2} & = & \frac{\left(\alpha+2\right)\left(\alpha+3\right)}{\alpha\left(\alpha+1\right)}-3\\
m_{d} & = & \textrm{NA}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Doubly Non-central F*
-\layout Section
+\end_layout
+\begin_layout Section
Doubly Non-central t*
-\layout Section
+\end_layout
+\begin_layout Section
Double Weibull
-\layout Standard
+\end_layout
+\begin_layout Standard
This is a signed form of the Weibull distribution.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{c}{2}\left|x\right|^{c-1}\exp\left(-\left|x\right|^{c}\right)\\
F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
@@ -1467,7 +1529,7 @@
-\log^{1/c}\left(\frac{1}{2q}\right) & & q\leq\frac{1}{2}\\
\log^{1/c}\left(\frac{1}{2q-1}\right) & & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
@@ -1475,7 +1537,7 @@
\Gamma\left(1+\frac{n}{c}\right) & n\textrm{ even}\\
0 & n\textrm{ odd}\end{cases}\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1485,34 +1547,38 @@
\gamma_{2} & = & \frac{\Gamma\left(1+\frac{4}{c}\right)}{\Gamma^{2}\left(1+\frac{2}{c}\right)}\\
m_{d} & = & \textrm{NA bimodal}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Erlang
-\layout Standard
+\end_layout
+\begin_layout Standard
This is just the Gamma distribution with shape parameter
\begin_inset Formula $\alpha=n$
-\end_inset
+\end_inset
an integer.
-\layout Section
+\end_layout
+\begin_layout Section
Exponential
-\layout Standard
+\end_layout
+\begin_layout Standard
This is a special case of the Gamma (and Erlang) distributions with shape
parameter
\begin_inset Formula $\left(\alpha=1\right)$
-\end_inset
+\end_inset
and the same location and scale parameters.
The standard form is therefore (
\begin_inset Formula $x\geq0$
-\end_inset
+\end_inset
)
\begin_inset Formula \begin{eqnarray*}
@@ -1520,25 +1586,25 @@
F\left(x\right) & = & \Gamma\left(1,x\right)=1-e^{-x}\\
G\left(q\right) & = & -\log\left(1-q\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=n!\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\frac{1}{1-t}\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1548,31 +1614,33 @@
\gamma_{2} & = & 6\\
m_{d} & = & 0\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=1.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Exponentiated Weibull
-\layout Standard
+\end_layout
+\begin_layout Standard
Two positive shape parameters
\begin_inset Formula $a$
-\end_inset
+\end_inset
and
\begin_inset Formula $c$
-\end_inset
+\end_inset
and
\begin_inset Formula $x\in\left(0,\infty\right)$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1580,22 +1648,24 @@
F\left(x;a,c\right) & = & \left[1-\exp\left(-x^{c}\right)\right]^{a}\\
G\left(q;a,c\right) & = & \left[-\log\left(1-q^{1/a}\right)\right]^{1/c}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Exponential Power
-\layout Standard
+\end_layout
+\begin_layout Standard
One positive shape parameter
\begin_inset Formula $b$
-\end_inset
+\end_inset
.
Defined for
\begin_inset Formula $x\geq0.$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1603,84 +1673,88 @@
F\left(x;b\right) & = & 1-\exp\left[1-e^{x^{b}}\right]\\
G\left(q;b\right) & = & \log^{1/b}\left[1-\log\left(1-q\right)\right]\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Fatigue Life (Birnbaum-Sanders)
-\layout Standard
+\end_layout
+\begin_layout Standard
This distribution's pdf is the average of the inverse-Gaussian
\begin_inset Formula $\left(\mu=1\right)$
-\end_inset
+\end_inset
and reciprocal inverse-Gaussian pdf
\begin_inset Formula $\left(\mu=1\right)$
-\end_inset
+\end_inset
.
We follow the notation of JKB here with
\begin_inset Formula $\beta=S.$
-\end_inset
+\end_inset
for
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{x+1}{2c\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-1\right)^{2}}{2xc^{2}}\right)\\
F\left(x;c\right) & = & \Phi\left(\frac{1}{c}\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)\right)\\
G\left(q;c\right) & = & \frac{1}{4}\left[c\Phi^{-1}\left(q\right)+\sqrt{c^{2}\left(\Phi^{-1}\left(q\right)\right)^{2}+4}\right]^{2}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=c\sqrt{2\pi}\exp\left[\frac{1}{c^{2}}\left(1-\sqrt{1-2c^{2}t}\right)\right]\left(1+\frac{1}{\sqrt{1-2c^{2}t}}\right)\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{c^{2}}{2}+1\\
\mu_{2} & = & c^{2}\left(\frac{5}{4}c^{2}+1\right)\\
\gamma_{1} & = & \frac{4c\sqrt{11c^{2}+6}}{\left(5c^{2}+4\right)^{3/2}}\\
\gamma_{2} & = & \frac{6c^{2}\left(93c^{2}+41\right)}{\left(5c^{2}+4\right)^{2}}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Fisk (Log Logistic)
-\layout Standard
+\end_layout
+\begin_layout Standard
Special case of the Burr distribution with
\begin_inset Formula $d=1$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
c & > & 0\\
k & = & \Gamma\left(1-\frac{2}{c}\right)\Gamma\left(\frac{2}{c}+1\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+1\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c,d\right) & = & \frac{cx^{c-1}}{\left(1+x^{c}\right)^{2}}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x;c,d\right) & = & \left(1+x^{-c}\right)^{-1}\\
@@ -1695,45 +1769,47 @@
m_{d} & = & \left(\frac{c-1}{c+1}\right)^{1/c}\,\textrm{if }c>1\,\textrm{otherwise }0\\
m_{n} & = & 1\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=2-\log c.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Folded Cauchy
-\layout Standard
+\end_layout
+\begin_layout Standard
This formula can be expressed in terms of the standard formulas for the
Cauchy distribution (call the cdf
\begin_inset Formula $C\left(x\right)$
-\end_inset
+\end_inset
and the pdf
\begin_inset Formula $d\left(x\right)$
-\end_inset
+\end_inset
).
if
\begin_inset Formula $Y$
-\end_inset
+\end_inset
is cauchy then
\begin_inset Formula $\left|Y\right|$
-\end_inset
+\end_inset
is folded cauchy.
Note that
\begin_inset Formula $x\geq0.$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1741,54 +1817,57 @@
F\left(x;c\right) & = & \frac{1}{\pi}\tan^{-1}\left(x-c\right)+\frac{1}{\pi}\tan^{-1}\left(x+c\right)\\
G\left(q;c\right) & = & F^{-1}\left(x;c\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
+\begin_layout Standard
No moments
-\layout Section
+\end_layout
+\begin_layout Section
Folded Normal
-\layout Standard
+\end_layout
+\begin_layout Standard
If
\begin_inset Formula $Z$
-\end_inset
+\end_inset
is Normal with mean
\begin_inset Formula $L$
-\end_inset
+\end_inset
and
\begin_inset Formula $\sigma=S$
-\end_inset
+\end_inset
, then
\begin_inset Formula $\left|Z\right|$
-\end_inset
+\end_inset
is a folded normal with shape parameter
\begin_inset Formula $c=\left|L\right|/S$
-\end_inset
+\end_inset
, location parameter
\begin_inset Formula $0$
-\end_inset
+\end_inset
and scale parameter
\begin_inset Formula $S$
-\end_inset
+\end_inset
.
This is a special case of the non-central chi distribution with one-degree
of freedom and non-centrality parameter
\begin_inset Formula $c^{2}.$
-\end_inset
+\end_inset
Note that
\begin_inset Formula $c\geq0$
-\end_inset
+\end_inset
.
The standard form of the folded normal is
@@ -1797,13 +1876,13 @@
F\left(x;c\right) & = & \Phi\left(x-c\right)-\Phi\left(-x-c\right)=\Phi\left(x-c\right)+\Phi\left(x+c\right)-1\\
G\left(\alpha;c\right) & = & F^{-1}\left(x;c\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\exp\left[\frac{t}{2}\left(t-2c\right)\right]\left(1+e^{2ct}\right)\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1814,38 +1893,40 @@
\gamma_{1} & = & \frac{\sqrt{\frac{2}{\pi}}p^{3}\left(4-\frac{\pi}{p^{2}}\left(2c^{2}+1\right)\right)+2ck\left(6p^{2}+3cpk\sqrt{2\pi}+\pi c\left(k^{2}-1\right)\right)}{\pi\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{c^{4}+6c^{2}+3+6\left(c^{2}+1\right)\mu^{2}-3\mu^{4}-4p\mu\left(\sqrt{\frac{2}{\pi}}\left(c^{2}+2\right)+\frac{ck}{p}\left(c^{2}+3\right)\right)}{\mu_{2}^{2}}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Fratio (or F)
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined for
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
.
The distribution of
\begin_inset Formula $\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right)$
-\end_inset
+\end_inset
if
\begin_inset Formula $X_{1}$
-\end_inset
+\end_inset
is chi-squared with
\begin_inset Formula $v_{1}$
-\end_inset
+\end_inset
degrees of freedom and
\begin_inset Formula $X_{2}$
-\end_inset
+\end_inset
is chi-squared with
\begin_inset Formula $v_{2}$
-\end_inset
+\end_inset
degrees of freedom.
@@ -1854,7 +1935,7 @@
F\left(x;v_{1},v_{2}\right) & = & I\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2},\frac{\nu_{2}x}{\nu_{2}+\nu_{1}x}\right)\\
G\left(q;\nu_{1},\nu_{2}\right) & = & \left[\frac{\nu_{2}}{I^{-1}\left(\nu_{1}/2,\nu_{2}/2,q\right)}-\frac{\nu_{1}}{\nu_{2}}\right]^{-1}.\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1863,22 +1944,24 @@
\gamma_{1} & = & \frac{2\left(2\nu_{1}+\nu_{2}-2\right)}{\nu_{2}-6}\sqrt{\frac{2\left(\nu_{2}-4\right)}{\nu_{1}\left(\nu_{1}+\nu_{2}-2\right)}}\quad\nu_{2}>6\\
\gamma_{2} & = & \frac{3\left[8+\left(\nu_{2}-6\right)\gamma_{1}^{2}\right]}{2\nu-16}\quad\nu_{2}>8\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Fréchet (ExtremeLB, Extreme Value II, Weibull minimum)
-\layout Standard
+\end_layout
+\begin_layout Standard
A type of extreme-value distribution with a lower bound.
Defined for
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1886,13 +1969,13 @@
F\left(x;c\right) & = & 1-\exp\left(-x^{c}\right)\\
G\left(q;c\right) & = & \left[-\log\left(1-q\right)\right]^{1/c}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=\Gamma\left(1+\frac{n}{c}\right)\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -1903,37 +1986,39 @@
m_{d} & = & \left(\frac{c}{1+c}\right)^{1/c}\\
m_{n} & = & G\left(\frac{1}{2};c\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
-\end_inset
+\end_inset
where
\begin_inset Formula $\gamma$
-\end_inset
+\end_inset
is Euler's constant and equal to
\begin_inset Formula \[
\gamma\approx0.57721566490153286061.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Fréchet (left-skewed, Extreme Value Type III, Weibull maximum)
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined for
\begin_inset Formula $x<0$
-\end_inset
+\end_inset
and
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
.
@@ -1942,53 +2027,56 @@
F\left(x;c\right) & = & \exp\left(-\left(-x\right)^{c}\right)\\
G\left(q;c\right) & = & -\left(-\log q\right)^{1/c}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
+\begin_layout Standard
The mean is the negative of the right-skewed Frechet distribution given
above, and the other statistical parameters can be computed from
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\left(-1\right)^{n}\Gamma\left(1+\frac{n}{c}\right).\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
-\end_inset
+\end_inset
where
\begin_inset Formula $\gamma$
-\end_inset
+\end_inset
is Euler's constant and equal to
\begin_inset Formula \[
\gamma\approx0.57721566490153286061.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Gamma
-\layout Standard
+\end_layout
+\begin_layout Standard
The standard form for the gamma distribution is
\begin_inset Formula $\left(\alpha>0\right)$
-\end_inset
+\end_inset
valid for
\begin_inset Formula $x\geq0$
-\end_inset
+\end_inset
.
\begin_inset Formula \begin{eqnarray*}
@@ -1996,13 +2084,13 @@
F\left(x;\alpha\right) & = & \Gamma\left(\alpha,x\right)\\
G\left(q;\alpha\right) & = & \Gamma^{-1}\left(\alpha,q\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\frac{1}{\left(1-t\right)^{\alpha}}\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2012,62 +2100,64 @@
\gamma_{2} & = & \frac{6}{\alpha}\\
m_{d} & = & \alpha-1\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\Psi\left(a\right)\left[1-a\right]+a+\log\Gamma\left(a\right)\]
-\end_inset
+\end_inset
where
\begin_inset Formula \[
\Psi\left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a\right)}.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Generalized Logistic
-\layout Standard
+\end_layout
+\begin_layout Standard
Has been used in the analysis of extreme values.
Has one shape parameter
\begin_inset Formula $c>0.$
-\end_inset
+\end_inset
And
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{c\exp\left(-x\right)}{\left[1+\exp\left(-x\right)\right]^{c+1}}\\
F\left(x;c\right) & = & \frac{1}{\left[1+\exp\left(-x\right)\right]^{c}}\\
G\left(q;c\right) & = & -\log\left(q^{-1/c}-1\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\frac{c}{1-t}\,_{2}F_{1}\left(1+c,\,1-t\,;\,2-t\,;-1\right)\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \gamma+\psi_{0}\left(c\right)\\
\mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\\
@@ -2076,7 +2166,7 @@
m_{d} & = & \log c\\
m_{n} & = & -\log\left(2^{1/c}-1\right)\end{eqnarray*}
-\end_inset
+\end_inset
Note that the polygamma function is
\begin_inset Formula \begin{eqnarray*}
@@ -2084,42 +2174,44 @@
& = & \left(-1\right)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}}\\
& = & \left(-1\right)^{n+1}n!\zeta\left(n+1,z\right)\end{eqnarray*}
-\end_inset
+\end_inset
where
\begin_inset Formula $\zeta\left(k,x\right)$
-\end_inset
+\end_inset
is a generalization of the Riemann zeta function called the Hurwitz zeta
function Note that
\begin_inset Formula $\zeta\left(n\right)\equiv\zeta\left(n,1\right)$
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Generalized Pareto
-\layout Standard
+\end_layout
+\begin_layout Standard
Shape parameter
\begin_inset Formula $c\neq0$
-\end_inset
+\end_inset
and defined for
\begin_inset Formula $x\geq0$
-\end_inset
+\end_inset
for all
\begin_inset Formula $c$
-\end_inset
+\end_inset
and
\begin_inset Formula $x<\frac{1}{\left|c\right|}$
-\end_inset
+\end_inset
if
\begin_inset Formula $c$
-\end_inset
+\end_inset
is negative.
@@ -2128,29 +2220,29 @@
F\left(x;c\right) & = & 1-\frac{1}{\left(1+cx\right)^{1/c}}\\
G\left(q;c\right) & = & \frac{1}{c}\left[\left(\frac{1}{1-q}\right)^{c}-1\right]\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\left\{ \begin{array}{cc}
\left(-\frac{t}{c}\right)^{\frac{1}{c}}e^{-\frac{t}{c}}\left[\Gamma\left(1-\frac{1}{c}\right)+\Gamma\left(-\frac{1}{c},-\frac{t}{c}\right)-\pi\csc\left(\frac{\pi}{c}\right)/\Gamma\left(\frac{1}{c}\right)\right] & c>0\\
\left(\frac{\left|c\right|}{t}\right)^{1/\left|c\right|}\Gamma\left[\frac{1}{\left|c\right|},\frac{t}{\left|c\right|}\right] & c<0\end{array}\right.\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{\left(-1\right)^{n}}{c^{n}}\sum_{k=0}^{n}\left(\begin{array}{c}
n\\
k\end{array}\right)\frac{\left(-1\right)^{k}}{1-ck}\quad cn<1\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2159,7 +2251,7 @@
\mu_{3}^{\prime} & = & \frac{6}{\left(1-c\right)\left(1-2c\right)\left(1-3c\right)}\quad c<\frac{1}{3}\\
\mu_{4}^{\prime} & = & \frac{24}{\left(1-c\right)\left(1-2c\right)\left(1-3c\right)\left(1-4c\right)}\quad c<\frac{1}{4}\end{eqnarray*}
-\end_inset
+\end_inset
Thus,
\begin_inset Formula \begin{eqnarray*}
@@ -2168,38 +2260,40 @@
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=1+c\quad c>0\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Generalized Exponential
-\layout Standard
+\end_layout
+\begin_layout Standard
Three positive shape parameters for
\begin_inset Formula $x\geq0.$
-\end_inset
+\end_inset
Note that
\begin_inset Formula $a,b,$
-\end_inset
+\end_inset
and
\begin_inset Formula $c$
-\end_inset
+\end_inset
are all
\begin_inset Formula $>0.$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2207,29 +2301,32 @@
F\left(x;a,b,c\right) & = & 1-\exp\left[ax-bx+\frac{b}{c}\left(1-e^{-cx}\right)\right]\\
G\left(q;a,b,c\right) & = & F^{-1}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Generalized Extreme Value
-\layout Standard
+\end_layout
+\begin_layout Standard
Extreme value distributions with shape parameter
\begin_inset Formula $c$
-\end_inset
+\end_inset
.
-\layout Standard
+\end_layout
+\begin_layout Standard
For
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
defined on
\begin_inset Formula $-\infty<x\leq1/c.$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2237,7 +2334,7 @@
F\left(x;c\right) & = & \exp\left[-\left(1-cx\right)^{1/c}\right]\\
G\left(q;c\right) & = & \frac{1}{c}\left[1-\left(-\log q\right)^{c}\right]\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
@@ -2245,7 +2342,7 @@
n\\
k\end{array}\right)\left(-1\right)^{k}\Gamma\left(ck+1\right)\quad cn>-1\]
-\end_inset
+\end_inset
So,
\begin_inset Formula \begin{eqnarray*}
@@ -2254,19 +2351,19 @@
\mu_{3}^{\prime} & = & \frac{1}{c^{3}}\left(1-3\Gamma\left(1+c\right)+3\Gamma\left(1+2c\right)-\Gamma\left(1+3c\right)\right)\quad c>-\frac{1}{3}\\
\mu_{4}^{\prime} & = & \frac{1}{c^{4}}\left(1-4\Gamma\left(1+c\right)+6\Gamma\left(1+2c\right)-4\Gamma\left(1+3c\right)+\Gamma\left(1+4c\right)\right)\quad c>-\frac{1}{4}\end{eqnarray*}
-\end_inset
+\end_inset
For
\begin_inset Formula $c<0$
-\end_inset
+\end_inset
defined on
\begin_inset Formula $\frac{1}{c}\leq x<\infty.$
-\end_inset
+\end_inset
For
\begin_inset Formula $c=0$
-\end_inset
+\end_inset
defined over all space
\begin_inset Formula \begin{eqnarray*}
@@ -2274,7 +2371,7 @@
F\left(x;0\right) & = & \exp\left[-e^{-x}\right]\\
G\left(q;0\right) & = & -\log\left(-\log q\right)\end{eqnarray*}
-\end_inset
+\end_inset
This is just the (left-skewed) Gumbel distribution for c=0.
@@ -2284,30 +2381,32 @@
\gamma_{1} & = & \frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\
\gamma_{2} & = & \frac{12}{5}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Generalized Gamma
-\layout Standard
+\end_layout
+\begin_layout Standard
A general probability form that reduces to many common distributions:
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
\begin_inset Formula $a>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $c\neq0.$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,c\right) & = & \frac{\left|c\right|x^{ca-1}}{\Gamma\left(a\right)}\exp\left(-x^{c}\right)\\
F\left(x;a,c\right) & = & \begin{array}{cc}
@@ -2316,13 +2415,13 @@
G\left(q;a,c\right) & = & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)q\right]\right\} ^{1/c}\quad c>0\\
& & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)\left(1-q\right)\right]\right\} ^{1/c}\quad c<0\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{\Gamma\left(a+\frac{n}{c}\right)}{\Gamma\left(a\right)}\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2332,47 +2431,49 @@
\gamma_{2} & = & \frac{\Gamma\left(a+\frac{4}{c}\right)/\Gamma\left(a\right)-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\\
m_{d} & = & \left(\frac{ac-1}{c}\right)^{1/c}.\end{eqnarray*}
-\end_inset
+\end_inset
Special cases are Weibull
\begin_inset Formula $\left(a=1\right)$
-\end_inset
+\end_inset
, half-normal
\begin_inset Formula $\left(a=1/2,c=2\right)$
-\end_inset
+\end_inset
and ordinary gamma distributions
\begin_inset Formula $c=1.$
-\end_inset
+\end_inset
If
\begin_inset Formula $c=-1$
-\end_inset
+\end_inset
then it is the inverted gamma distribution.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=a-a\Psi\left(a\right)+\frac{1}{c}\Psi\left(a\right)+\log\Gamma\left(a\right)-\log\left|c\right|.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Generalized Half-Logistic
-\layout Standard
+\end_layout
+\begin_layout Standard
For
\begin_inset Formula $x\in\left[0,1/c\right]$
-\end_inset
+\end_inset
and
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
we have
\begin_inset Formula \begin{eqnarray*}
@@ -2380,34 +2481,36 @@
F\left(x;c\right) & = & \frac{1-\left(1-cx\right)^{1/c}}{1+\left(1-cx\right)^{1/c}}\\
G\left(q;c\right) & = & \frac{1}{c}\left[1-\left(\frac{1-q}{1+q}\right)^{c}\right]\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & 2-\left(2c+1\right)\log2.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Gilbrat
-\layout Standard
+\end_layout
+\begin_layout Standard
Special case of the log-normal with
\begin_inset Formula $\sigma=1$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=1.0$
-\end_inset
+\end_inset
(typically also
\begin_inset Formula $L=0.0$
-\end_inset
+\end_inset
)
\begin_inset Formula \begin{eqnarray*}
@@ -2415,7 +2518,7 @@
F\left(x;\sigma\right) & = & \Phi\left(\log x\right)=\frac{1}{2}\left[1+\textrm{erf}\left(\frac{\log x}{\sqrt{2}}\right)\right]\\
G\left(q;\sigma\right) & = & \exp\left\{ \Phi^{-1}\left(q\right)\right\} \end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2424,58 +2527,60 @@
\gamma_{1} & = & \sqrt{e-1}\left(2+e\right)\\
\gamma_{2} & = & e^{4}+2e^{3}+3e^{2}-6\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(\sqrt{2\pi e}\right)\\
& \approx & 1.4189385332046727418\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Gompertz (Truncated Gumbel)
-\layout Standard
+\end_layout
+\begin_layout Standard
For
\begin_inset Formula $x\geq0$
-\end_inset
+\end_inset
and
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
.
In JKB the two shape parameters
\begin_inset Formula $b,a$
-\end_inset
+\end_inset
are reduced to the single shape-parameter
\begin_inset Formula $c=b/a$
-\end_inset
+\end_inset
.
As
\begin_inset Formula $a$
-\end_inset
+\end_inset
is just a scale parameter when
\begin_inset Formula $a\neq0$
-\end_inset
+\end_inset
.
If
\begin_inset Formula $a=0,$
-\end_inset
+\end_inset
the distribution reduces to the exponential distribution scaled by
\begin_inset Formula $1/b.$
-\end_inset
+\end_inset
Thus, the standard form is given as
\begin_inset Formula \begin{eqnarray*}
@@ -2483,45 +2588,47 @@
F\left(x;c\right) & = & 1-\exp\left[-c\left(e^{x}-1\right)\right]\\
G\left(q;c\right) & = & \log\left[1-\frac{1}{c}\log\left(1-q\right)\right]\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=1-\log\left(c\right)-e^{c}\textrm{Ei}\left(1,c\right),\]
-\end_inset
+\end_inset
where
\begin_inset Formula \[
\textrm{Ei}\left(n,x\right)=\int_{1}^{\infty}t^{-n}\exp\left(-xt\right)dt\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value)
-\layout Standard
+\end_layout
+\begin_layout Standard
One of a clase of extreme value distributions (right-skewed).
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \exp\left(-\left(x+e^{-x}\right)\right)\\
F\left(x\right) & = & \exp\left(-e^{-x}\right)\\
G\left(q\right) & = & -\log\left(-\log\left(q\right)\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\Gamma\left(1-t\right)\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2532,80 +2639,83 @@
m_{d} & = & 0\\
m_{n} & = & -\log\left(\log2\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]\approx1.0608407169541684911\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Gumbel Left-skewed (for minimum order statistic)
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \exp\left(x-e^{x}\right)\\
F\left(x\right) & = & 1-\exp\left(-e^{x}\right)\\
G\left(q\right) & = & \log\left(-\log\left(1-q\right)\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\Gamma\left(1+t\right)\]
-\end_inset
+\end_inset
Note, that
\begin_inset Formula $\mu$
-\end_inset
+\end_inset
is negative the mean for the right-skewed distribution.
Similar for median and mode.
All other moments are the same.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]\approx1.0608407169541684911.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
HalfCauchy
-\layout Standard
+\end_layout
+\begin_layout Standard
If
\begin_inset Formula $Z$
-\end_inset
+\end_inset
is Hyperbolic Secant distributed then
\begin_inset Formula $e^{Z}$
-\end_inset
+\end_inset
is Half-Cauchy distributed.
Also, if
\begin_inset Formula $W$
-\end_inset
+\end_inset
is (standard) Cauchy distributed, then
\begin_inset Formula $\left|W\right|$
-\end_inset
+\end_inset
is Half-Cauchy distributed.
Special case of the Folded Cauchy distribution with
\begin_inset Formula $c=0.$
-\end_inset
+\end_inset
The standard form is
\begin_inset Formula \begin{eqnarray*}
@@ -2613,68 +2723,70 @@
F\left(x\right) & = & \frac{2}{\pi}\arctan\left(x\right)I_{\left[0,\infty\right]}\left(x\right)\\
G\left(q\right) & = & \tan\left(\frac{\pi}{2}q\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\cos t+\frac{2}{\pi}\left[\textrm{Si}\left(t\right)\cos t-\textrm{Ci}\left(\textrm{-}t\right)\sin t\right]\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
m_{d} & = & 0\\
m_{n} & = & \tan\left(\frac{\pi}{4}\right)\end{eqnarray*}
-\end_inset
+\end_inset
No moments, as the integrals diverge.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(2\pi\right)\\
& \approx & 1.8378770664093454836.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
HalfNormal
-\layout Standard
+\end_layout
+\begin_layout Standard
This is a special case of the chi distribution with
\begin_inset Formula $L=a$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=b$
-\end_inset
+\end_inset
and
\begin_inset Formula $\nu=1.$
-\end_inset
+\end_inset
This is also a special case of the folded normal with shape parameter
\begin_inset Formula $c=0$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=S.$
-\end_inset
+\end_inset
If
\begin_inset Formula $Z$
-\end_inset
+\end_inset
is (standard) normally distributed then,
\begin_inset Formula $\left|Z\right|$
-\end_inset
+\end_inset
is half-normal.
The standard form is
@@ -2683,18 +2795,18 @@
F\left(x\right) & = & 2\Phi\left(x\right)-1\\
G\left(q\right) & = & \Phi^{-1}\left(\frac{1+q}{2}\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\sqrt{2\pi}e^{t^{2}/2}\Phi\left(t\right)\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \sqrt{\frac{2}{\pi}}\\
\mu_{2} & = & 1-\frac{2}{\pi}\\
@@ -2703,40 +2815,42 @@
m_{d} & = & 0\\
m_{n} & = & \Phi^{-1}\left(\frac{3}{4}\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(\sqrt{\frac{\pi e}{2}}\right)\\
& \approx & 0.72579135264472743239.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Half-Logistic
-\layout Standard
+\end_layout
+\begin_layout Standard
In the limit as
\begin_inset Formula $c\rightarrow\infty$
-\end_inset
+\end_inset
for the generalized half-logistic we have the half-logistic defined over
\begin_inset Formula $x\geq0.$
-\end_inset
+\end_inset
Also, the distribution of
\begin_inset Formula $\left|X\right|$
-\end_inset
+\end_inset
where
\begin_inset Formula $X$
-\end_inset
+\end_inset
has logistic distribtution.
@@ -2745,22 +2859,22 @@
F\left(x\right) & = & \frac{1-e^{-x}}{1+e^{-x}}=\tanh\left(\frac{x}{2}\right)\\
G\left(q\right) & = & \log\left(\frac{1+q}{1-q}\right)=2\textrm{arctanh}\left(q\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=1-t\psi_{0}\left(\frac{1}{2}-\frac{t}{2}\right)+t\psi_{0}\left(1-\frac{t}{2}\right)\]
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=2\left(1-2^{1-n}\right)n!\zeta\left(n\right)\quad n\neq1\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2769,28 +2883,30 @@
\mu_{3}^{\prime} & = & 9\zeta\left(3\right)\\
\mu_{4}^{\prime} & = & 42\zeta\left(4\right)=\frac{7\pi^{4}}{15}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & 2-\log\left(2\right)\\
& \approx & 1.3068528194400546906.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Hyperbolic Secant
-\layout Standard
+\end_layout
+\begin_layout Standard
Related to the logistic distribution and used in lifetime analysis.
Standard form is (defined over all
\begin_inset Formula $x$
-\end_inset
+\end_inset
)
\begin_inset Formula \begin{eqnarray*}
@@ -2798,13 +2914,13 @@
F\left(x\right) & = & \frac{2}{\pi}\arctan\left(e^{x}\right)\\
G\left(q\right) & = & \log\left(\tan\left(\frac{\pi}{2}q\right)\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
M\left(t\right)=\sec\left(\frac{\pi}{2}t\right)\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2813,30 +2929,30 @@
0 & n\textrm{ odd}\\
C_{n/2}\frac{\pi^{n}}{2^{n}} & n\textrm{ even}\end{array}\right.\end{eqnarray*}
-\end_inset
+\end_inset
where
\begin_inset Formula $C_{m}$
-\end_inset
+\end_inset
is an integer given by
\begin_inset Formula \begin{eqnarray*}
C_{m} & = & \frac{\left(2m\right)!\left[\zeta\left(2m+1,\frac{1}{4}\right)-\zeta\left(2m+1,\frac{3}{4}\right)\right]}{\pi^{2m+1}2^{2m}}\\
& = & 4\left(-1\right)^{m-1}\frac{16^{m}}{2m+1}B_{2m+1}\left(\frac{1}{4}\right)\end{eqnarray*}
-\end_inset
+\end_inset
where
\begin_inset Formula $B_{2m+1}\left(\frac{1}{4}\right)$
-\end_inset
+\end_inset
is the Bernoulli polynomial of order
\begin_inset Formula $2m+1$
-\end_inset
+\end_inset
evaluated at
\begin_inset Formula $1/4.$
-\end_inset
+\end_inset
Thus
\begin_inset Formula \[
@@ -2844,90 +2960,93 @@
0 & n\textrm{ odd}\\
4\left(-1\right)^{n/2-1}\frac{\left(2\pi\right)^{n}}{n+1}B_{n+1}\left(\frac{1}{4}\right) & n\textrm{ even}\end{array}\right.\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
m_{d}=m_{n}=\mu & = & 0\\
\mu_{2} & = & \frac{\pi^{2}}{4}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & 2\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\log\left(2\pi\right).\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Gauss Hypergeometric
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula $x\in\left[0,1\right]$
-\end_inset
+\end_inset
,
\begin_inset Formula $\alpha>0,\,\beta>0$
-\end_inset
+\end_inset
\begin_inset Formula \[
C^{-1}=B\left(\alpha,\beta\right)\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha,\beta,\gamma,z\right) & = & Cx^{\alpha-1}\frac{\left(1-x\right)^{\beta-1}}{\left(1+zx\right)^{\gamma}}\\
\mu_{n}^{\prime} & = & \frac{B\left(n+\alpha,\beta\right)}{B\left(\alpha,\beta\right)}\frac{\,_{2}F_{1}\left(\gamma,\alpha+n;\alpha+\beta+n;-z\right)}{\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Inverted Gamma
-\layout Standard
+\end_layout
+\begin_layout Standard
Special case of the generalized Gamma distribution with
\begin_inset Formula $c=-1$
-\end_inset
+\end_inset
and
\begin_inset Formula $a>0$
-\end_inset
+\end_inset
,
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;a\right) & = & \frac{x^{-a-1}}{\Gamma\left(a\right)}\exp\left(-\frac{1}{x}\right)\\
F\left(x;a\right) & = & \frac{\Gamma\left(a,\frac{1}{x}\right)}{\Gamma\left(a\right)}\\
G\left(q;a\right) & = & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)q\right]\right\} ^{-1}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{\Gamma\left(a-n\right)}{\Gamma\left(a\right)}\quad a>n\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2936,54 +3055,56 @@
\gamma_{1} & = & \frac{\frac{1}{\left(a-3\right)\left(a-2\right)\left(a-1\right)}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\frac{1}{\left(a-4\right)\left(a-3\right)\left(a-2\right)\left(a-1\right)}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
m_{d}=\frac{1}{a+1}\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=a-\left(a+1\right)\Psi\left(a\right)+\log\Gamma\left(a\right).\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Inverse Normal (Inverse Gaussian)
-\layout Standard
+\end_layout
+\begin_layout Standard
The standard form involves the shape parameter
\begin_inset Formula $\mu$
-\end_inset
+\end_inset
(in most definitions,
\begin_inset Formula $L=0.0$
-\end_inset
+\end_inset
is used).
(In terms of the regress documentation
\begin_inset Formula $\mu=A/B$
-\end_inset
+\end_inset
) and
\begin_inset Formula $B=S$
-\end_inset
+\end_inset
and
\begin_inset Formula $L$
-\end_inset
+\end_inset
is not a parameter in that distribution.
A standard form is
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -2991,12 +3112,12 @@
F\left(x;\mu\right) & = & \Phi\left(\frac{1}{\sqrt{x}}\frac{x-\mu}{\mu}\right)+\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{x+\mu}{\mu}\right)\\
G\left(q;\mu\right) & = & F^{-1}\left(q;\mu\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \mu\\
\mu_{2} & = & \mu^{3}\\
@@ -3004,80 +3125,83 @@
\gamma_{2} & = & 15\mu\\
m_{d} & = & \frac{\mu}{2}\left(\sqrt{9\mu^{2}+4}-3\mu\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
+\begin_layout Standard
This is related to the canonical form or JKB
\begin_inset Quotes eld
-\end_inset
+\end_inset
two-parameter
\begin_inset Quotes erd
-\end_inset
+\end_inset
inverse Gaussian when written in it's full form with scale parameter
\begin_inset Formula $S$
-\end_inset
+\end_inset
and location parameter
\begin_inset Formula $L$
-\end_inset
+\end_inset
by taking
\begin_inset Formula $L=0$
-\end_inset
+\end_inset
and
\begin_inset Formula $S\equiv\lambda,$
-\end_inset
+\end_inset
then
\begin_inset Formula $\mu S$
-\end_inset
+\end_inset
is equal to
\begin_inset Formula $\mu_{2}$
-\end_inset
+\end_inset
where
-\bar under
+\bar under
-\bar default
+\bar default
\begin_inset Formula $\mu_{2}$
-\end_inset
+\end_inset
is the parameter used by JKB.
We prefer this form because of it's consistent use of the scale parameter.
Notice that in JKB the skew
\begin_inset Formula $\left(\sqrt{\beta_{1}}\right)$
-\end_inset
+\end_inset
and the kurtosis (
\begin_inset Formula $\beta_{2}-3$
-\end_inset
+\end_inset
) are both functions only of
\begin_inset Formula $\mu_{2}/\lambda=\mu S/S=\mu$
-\end_inset
+\end_inset
as shown here, while the variance and mean of the standard form here are
transformed appropriately.
-\layout Section
+\end_layout
+\begin_layout Section
Inverted Weibull
-\layout Standard
+\end_layout
+\begin_layout Standard
Shape parameter
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
.
Then
@@ -3086,35 +3210,37 @@
F\left(x;c\right) & = & \exp\left(-x^{-c}\right)\\
G\left(q;c\right) & = & \left(-\log q\right)^{-1/c}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=1+\gamma+\frac{\gamma}{c}-\log\left(c\right)\]
-\end_inset
+\end_inset
where
\begin_inset Formula $\gamma$
-\end_inset
+\end_inset
is Euler's constant.
-\layout Section
+\end_layout
+\begin_layout Section
Johnson SB
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined for
\begin_inset Formula $x\in\left(0,1\right)$
-\end_inset
+\end_inset
with two shape parameters
\begin_inset Formula $a$
-\end_inset
+\end_inset
and
\begin_inset Formula $b>0.$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3122,25 +3248,27 @@
F\left(x;a,b\right) & = & \Phi\left(a+b\log\frac{x}{1-x}\right)\\
G\left(q;a,b\right) & = & \frac{1}{1+\exp\left[-\frac{1}{b}\left(\Phi^{-1}\left(q\right)-a\right)\right]}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Johnson SU
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined for all
\begin_inset Formula $x$
-\end_inset
+\end_inset
with two shape parameters
\begin_inset Formula $a$
-\end_inset
+\end_inset
and
\begin_inset Formula $b>0$
-\end_inset
+\end_inset
.
@@ -3149,21 +3277,24 @@
F\left(x;a,b\right) & = & \Phi\left(a+b\log\left(x+\sqrt{x^{2}+1}\right)\right)\\
G\left(q;a,b\right) & = & \sinh\left[\frac{\Phi^{-1}\left(q\right)-a}{b}\right]\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
KSone
-\layout Section
+\end_layout
+\begin_layout Section
KStwo
-\layout Section
+\end_layout
+\begin_layout Section
Laplace (Double Exponential, Bilateral Expoooonential)
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{2}e^{-\left|x\right|}\\
F\left(x\right) & = & \left\{ \begin{array}{ccc}
@@ -3173,7 +3304,7 @@
\log\left(2q\right) & & q\leq\frac{1}{2}\\
-\log\left(2-2q\right) & & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3182,95 +3313,98 @@
\gamma_{1} & = & 0\\
\gamma_{2} & = & 3\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
+\begin_layout Standard
The ML estimator of the location parameter is
\begin_inset Formula \[
\hat{L}=\textrm{median}\left(X_{i}\right)\]
-\end_inset
+\end_inset
where
\begin_inset Formula $X_{i}$
-\end_inset
+\end_inset
is a sequence of
\begin_inset Formula $N$
-\end_inset
+\end_inset
mutually independent Laplace RV's and the median is some number between
the
\begin_inset Formula $\frac{1}{2}N\textrm{th}$
-\end_inset
+\end_inset
and the
\begin_inset Formula $(N/2+1)\textrm{th}$
-\end_inset
+\end_inset
order statistic (
-\emph on
+\emph on
e.g.
-\emph default
+\emph default
take the average of these two) when
\begin_inset Formula $N$
-\end_inset
+\end_inset
is even.
Also,
\begin_inset Formula \[
\hat{S}=\frac{1}{N}\sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|.\]
-\end_inset
+\end_inset
Replace
\begin_inset Formula $\hat{L}$
-\end_inset
+\end_inset
with
\begin_inset Formula $L$
-\end_inset
+\end_inset
if it is known.
If
\begin_inset Formula $L$
-\end_inset
+\end_inset
is known then this estimator is distributed as
\begin_inset Formula $\left(2N\right)^{-1}S\cdot\chi_{2N}^{2}$
-\end_inset
+\end_inset
.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(2e\right)\\
& \approx & 1.6931471805599453094.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Left-skewed Lévy
-\layout Standard
+\end_layout
+\begin_layout Standard
Special case of Lévy-stable distribution with
\begin_inset Formula $\alpha=\frac{1}{2}$
-\end_inset
+\end_inset
and
\begin_inset Formula $\beta=-1$
-\end_inset
+\end_inset
the support is
\begin_inset Formula $x<0$
-\end_inset
+\end_inset
.
In standard form
@@ -3279,26 +3413,28 @@
F\left(x\right) & = & 2\Phi\left(\frac{1}{\sqrt{\left|x\right|}}\right)-1\\
G\left(q\right) & = & -\left[\Phi^{-1}\left(\frac{q+1}{2}\right)\right]^{-2}.\end{eqnarray*}
-\end_inset
+\end_inset
No moments.
-\layout Section
+\end_layout
+\begin_layout Section
Lévy
-\layout Standard
+\end_layout
+\begin_layout Standard
A special case of Lévy-stable distributions with
\begin_inset Formula $\alpha=\frac{1}{2}$
-\end_inset
+\end_inset
and
\begin_inset Formula $\beta=1$
-\end_inset
+\end_inset
.
In standard form it is defined for
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
as
\begin_inset Formula \begin{eqnarray*}
@@ -3306,37 +3442,39 @@
F\left(x\right) & = & 2\left[1-\Phi\left(\frac{1}{\sqrt{x}}\right)\right]\\
G\left(q\right) & = & \left[\Phi^{-1}\left(1-\frac{q}{2}\right)\right]^{-2}.\end{eqnarray*}
-\end_inset
+\end_inset
It has no finite moments.
-\layout Section
+\end_layout
+\begin_layout Section
Logistic (Sech-squared)
-\layout Standard
+\end_layout
+\begin_layout Standard
A special case of the Generalized Logistic distribution with
\begin_inset Formula $c=1.$
-\end_inset
+\end_inset
Defined for
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{\exp\left(-x\right)}{\left[1+\exp\left(-x\right)\right]^{2}}\\
F\left(x\right) & = & \frac{1}{1+\exp\left(-x\right)}\\
G\left(q\right) & = & -\log\left(1/q-1\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \gamma+\psi_{0}\left(1\right)=0\\
\mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(1\right)=\frac{\pi^{2}}{3}\\
@@ -3345,30 +3483,32 @@
m_{d} & = & \log1=0\\
m_{n} & = & -\log\left(2-1\right)=0\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=1.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Log Double Exponential (Log-Laplace)
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined over
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
with
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3382,30 +3522,32 @@
\left(2q\right)^{1/c} & & 0\leq q<\frac{1}{2}\\
\left(2-2q\right)^{-1/c} & & \frac{1}{2}\leq q\leq1\end{array}\right.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\log\left(\frac{2e}{c}\right)\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Log Gamma
-\layout Standard
+\end_layout
+\begin_layout Standard
A single shape parameter
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
(Defined for all
\begin_inset Formula $x$
-\end_inset
+\end_inset
)
\begin_inset Formula \begin{eqnarray*}
@@ -3413,13 +3555,13 @@
F\left(x;c\right) & = & \frac{\Gamma\left(c,e^{x}\right)}{\Gamma\left(c\right)}\\
G\left(q;c\right) & = & \log\left[\Gamma^{-1}\left[c,q\Gamma\left(c\right)\right]\right]\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=\int_{0}^{\infty}\left[\log y\right]^{n}y^{c-1}\exp\left(-y\right)dy.\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3428,43 +3570,45 @@
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Log Normal (Cobb-Douglass)
-\layout Standard
+\end_layout
+\begin_layout Standard
Has one shape parameter
\begin_inset Formula $\sigma$
-\end_inset
+\end_inset
>0.
(Notice that the
\begin_inset Quotes eld
-\end_inset
+\end_inset
Regress
\begin_inset Quotes erd
-\end_inset
+\end_inset
\begin_inset Formula $A=\log S$
-\end_inset
+\end_inset
where
\begin_inset Formula $S$
-\end_inset
+\end_inset
is the scale parameter and
\begin_inset Formula $A$
-\end_inset
+\end_inset
is the mean of the underlying normal distribution).
The standard form is
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3472,7 +3616,7 @@
F\left(x;\sigma\right) & = & \Phi\left(\frac{\log x}{\sigma}\right)\\
G\left(q;\sigma\right) & = & \exp\left\{ \sigma\Phi^{-1}\left(q\right)\right\} \end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3481,45 +3625,80 @@
\gamma_{1} & = & \sqrt{p-1}\left(2+p\right)\\
\gamma_{2} & = & p^{4}+2p^{3}+3p^{2}-6\quad\quad p=e^{\sigma^{2}}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
+\begin_layout Standard
Notice that using JKB notation we have
\begin_inset Formula $\theta=L,$
-\end_inset
+\end_inset
\begin_inset Formula $\zeta=\log S$
-\end_inset
+\end_inset
and we have given the so-called antilognormal form of the distribution.
This is more consistent with the location, scale parameter description
of general probability distributions.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\frac{1}{2}\left[1+\log\left(2\pi\right)+2\log\left(\sigma\right)\right].\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Standard
+Also, note that if
+\begin_inset Formula $X$
+\end_inset
+
+ is a log-normally distributed random-variable with
+\begin_inset Formula $L=0$
+\end_inset
+
+ and
+\begin_inset Formula $S$
+\end_inset
+
+ and shape parameter
+\begin_inset Formula $\sigma.$
+\end_inset
+
+ Then,
+\begin_inset Formula $\log X$
+\end_inset
+
+ is normally distributed with variance
+\begin_inset Formula $\sigma^{2}$
+\end_inset
+
+ and mean
+\begin_inset Formula $\log S.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
Nakagami
-\layout Standard
+\end_layout
+\begin_layout Standard
Generalization of the chi distribution.
Shape parameter is
\begin_inset Formula $\nu>0.$
-\end_inset
+\end_inset
Defined for
\begin_inset Formula $x>0.$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3527,7 +3706,7 @@
F\left(x;\nu\right) & = & \Gamma\left(\nu,\nu x^{2}\right)\\
G\left(q;\nu\right) & = & \sqrt{\frac{1}{\nu}\Gamma^{-1}\left(v,q\right)}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3536,70 +3715,75 @@
\gamma_{1} & = & \frac{\mu\left(1-4v\mu_{2}\right)}{2\nu\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{-6\mu^{4}\nu+\left(8\nu-2\right)\mu^{2}-2\nu+1}{\nu\mu_{2}^{2}}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Noncentral beta*
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined over
\begin_inset Formula $x\in\left[0,1\right]$
-\end_inset
+\end_inset
with
\begin_inset Formula $a>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $b>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $c\geq0$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
F\left(x;a,b,c\right)=\sum_{j=0}^{\infty}\frac{e^{-c/2}\left(\frac{c}{2}\right)^{j}}{j!}I_{B}\left(a+j,b;0\right)\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Noncentral chi*
-\layout Section
+\end_layout
+\begin_layout Section
Noncentral chi-squared
-\layout Standard
+\end_layout
+\begin_layout Standard
The distribution of
\begin_inset Formula $\sum_{i=1}^{\nu}\left(Z_{i}+\delta_{i}\right)^{2}$
-\end_inset
+\end_inset
where
\begin_inset Formula $Z_{i}$
-\end_inset
+\end_inset
are independent standard normal variables and
\begin_inset Formula $\delta_{i}$
-\end_inset
+\end_inset
are constants.
\begin_inset Formula $\lambda=\sum_{i=1}^{\nu}\delta_{i}^{2}>0.$
-\end_inset
+\end_inset
(In communications it is called the Marcum-Q function).
Can be thought of as a Generalized Rayleigh-Rice distribution.
For
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3607,7 +3791,7 @@
F\left(x;\nu,\lambda\right) & = & \sum_{j=0}^{\infty}\left\{ \frac{\left(\lambda/2\right)^{j}}{j!}e^{-\lambda/2}\right\} \textrm{Pr}\left[\chi_{\nu+2j}^{2}\leq x\right]\\
G\left(q;\nu,\lambda\right) & = & F^{-1}\left(x;\nu,\lambda\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3616,68 +3800,72 @@
\gamma_{1} & = & \frac{\sqrt{8}\left(\nu+3\lambda\right)}{\left(\nu+2\lambda\right)^{3/2}}\\
\gamma_{2} & = & \frac{12\left(\nu+4\lambda\right)}{\left(\nu+2\lambda\right)^{2}}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Noncentral F
-\layout Standard
+\end_layout
+\begin_layout Standard
Let
\begin_inset Formula $\lambda>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $\nu_{1}>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $\nu_{2}>0.$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\lambda,\nu_{1},\nu_{2}\right) & = & \exp\left[\frac{\lambda}{2}+\frac{\left(\lambda\nu_{1}x\right)}{2\left(\nu_{1}x+\nu_{2}\right)}\right]\nu_{1}^{\nu_{1}/2}\nu_{2}^{\nu_{2}/2}x^{\nu_{1}/2-1}\\
& & \times\left(\nu_{2}+\nu_{1}x\right)^{-\left(\nu_{1}+\nu_{2}\right)/2}\frac{\Gamma\left(\frac{\nu_{1}}{2}\right)\Gamma\left(1+\frac{\nu_{2}}{2}\right)L_{\nu_{2}/2}^{\nu_{1}/2-1}\left(-\frac{\lambda\nu_{1}x}{2\left(\nu_{1}x+\nu_{2}\right)}\right)}{B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)\Gamma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Noncentral t
-\layout Standard
+\end_layout
+\begin_layout Standard
The distribution of the ratio
\begin_inset Formula \[
\frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}}\]
-\end_inset
+\end_inset
where
\begin_inset Formula $U$
-\end_inset
+\end_inset
and
\begin_inset Formula $\chi_{\nu}$
-\end_inset
+\end_inset
are independent and distributed as a standard normal and chi with
\begin_inset Formula $\nu$
-\end_inset
+\end_inset
degrees of freedom.
Note
\begin_inset Formula $\lambda>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $\nu>0$
-\end_inset
+\end_inset
.
@@ -3689,77 +3877,81 @@
& & \times\left(\frac{\nu}{\nu+x^{2}}\right)^{\left(\nu-1\right)/2}Hh_{\nu}\left(-\frac{\lambda x}{\sqrt{\nu+x^{2}}}\right)\\
F\left(x;\lambda,\nu\right) & =\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Normal
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{e^{-x^{2}/2}}{\sqrt{2\pi}}\\
F\left(x\right) & = & \Phi\left(x\right)=\frac{1}{2}+\frac{1}{2}\textrm{erf}\left(\frac{\textrm{x}}{\sqrt{2}}\right)\\
G\left(q\right) & = & \Phi^{-1}\left(q\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
-\align center
+\end_layout
+\begin_layout Standard
+\align center
\begin_inset Formula \begin{eqnarray*}
m_{d}=m_{n}=\mu & = & 0\\
\mu_{2} & = & 1\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & 0\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(\sqrt{2\pi e}\right)\\
& \approx & 1.4189385332046727418\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Maxwell
-\layout Standard
+\end_layout
+\begin_layout Standard
This is a special case of the Chi distribution with
\begin_inset Formula $L=0$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=S=\frac{1}{\sqrt{a}}$
-\end_inset
+\end_inset
and
\begin_inset Formula $\nu=3.$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \sqrt{\frac{2}{\pi}}x^{2}e^{-x^{2}/2}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x\right) & = & \Gamma\left(\frac{3}{2},\frac{x^{2}}{2}\right)\\
G\left(\alpha\right) & = & \sqrt{2\Gamma^{-1}\left(\frac{3}{2},\alpha\right)}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & 2\sqrt{\frac{2}{\pi}}\\
\mu_{2} & = & 3-\frac{8}{\pi}\\
@@ -3768,37 +3960,39 @@
m_{d} & = & \sqrt{2}\\
m_{n} & = & \sqrt{2\Gamma^{-1}\left(\frac{3}{2},\frac{1}{2}\right)}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=\log\left(\sqrt{\frac{2\pi}{e}}\right)+\gamma.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Mielke's Beta-Kappa
-\layout Standard
+\end_layout
+\begin_layout Standard
A generalized F distribution.
Two shape parameters
\begin_inset Formula $\kappa$
-\end_inset
+\end_inset
and
\begin_inset Formula $\theta$
-\end_inset
+\end_inset
, and
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
.
The
\begin_inset Formula $\beta$
-\end_inset
+\end_inset
in the DATAPLOT reference is a scale parameter.
\begin_inset Formula \begin{eqnarray*}
@@ -3806,107 +4000,112 @@
F\left(x;\kappa,\theta\right) & = & \frac{x^{\kappa}}{\left(1+x^{\theta}\right)^{\kappa/\theta}}\\
G\left(q;\kappa,\theta\right) & = & \left(\frac{q^{\theta/\kappa}}{1-q^{\theta/\kappa}}\right)^{1/\theta}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Pareto
-\layout Standard
+\end_layout
+\begin_layout Standard
For
\begin_inset Formula $x\geq1$
-\end_inset
+\end_inset
and
\begin_inset Formula $b>0$
-\end_inset
+\end_inset
.
Standard form is
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;b\right) & = & \frac{b}{x^{b+1}}\\
F\left(x;b\right) & = & 1-\frac{1}{x^{b}}\\
G\left(q;b\right) & = & \left(1-q\right)^{-1/b}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{b}{b-1}\quad b>1\\
\mu_{2} & = & \frac{b}{\left(b-2\right)\left(b-1\right)^{2}}\quad b>2\\
\gamma_{1} & = & \frac{2\left(b+1\right)\sqrt{b-2}}{\left(b-3\right)\sqrt{b}}\quad b>3\\
\gamma_{2} & = & \frac{6\left(b^{3}+b^{2}-6b-2\right)}{b\left(b^{2}-7b+12\right)}\quad b>4\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left(X\right)=\frac{1}{c}+1-\log\left(c\right)\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Pareto Second Kind (Lomax)
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula $c>0.$
-\end_inset
+\end_inset
This is Pareto of the first kind with
\begin_inset Formula $L=-1.0$
-\end_inset
+\end_inset
so
\begin_inset Formula $x\geq0$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{c}{\left(1+x\right)^{c+1}}\\
F\left(x;c\right) & = & 1-\frac{1}{\left(1+x\right)^{c}}\\
G\left(q;c\right) & = & \left(1-q\right)^{-1/c}-1\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=\frac{1}{c}+1-\log\left(c\right).\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Power Log Normal
-\layout Standard
+\end_layout
+\begin_layout Standard
A generalization of the log-normal distribution
\begin_inset Formula $\sigma>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3914,13 +4113,13 @@
F\left(x;\sigma,c\right) & = & 1-\left(\Phi\left(-\frac{\log x}{\sigma}\right)\right)^{c}\\
G\left(q;\sigma,c\right) & = & \exp\left[-\sigma\Phi^{-1}\left[\left(1-q\right)^{1/c}\right]\right]\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=\int_{0}^{1}\exp\left[-n\sigma\Phi^{-1}\left(y^{1/c}\right)\right]dy\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3929,21 +4128,23 @@
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
-\end_inset
+\end_inset
This distribution reduces to the log-normal distribution when
\begin_inset Formula $c=1.$
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Power Normal
-\layout Standard
+\end_layout
+\begin_layout Standard
A generalization of the normal distribution,
\begin_inset Formula $c>0$
-\end_inset
+\end_inset
for
\begin_inset Formula \begin{eqnarray*}
@@ -3951,13 +4152,13 @@
F\left(x;c\right) & = & 1-\left(\Phi\left(-x\right)\right)^{c}\\
G\left(q;c\right) & = & -\Phi^{-1}\left[\left(1-q\right)^{1/c}\right]\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=\left(-1\right)^{n}\int_{0}^{1}\left[\Phi^{-1}\left(y^{1/c}\right)\right]^{n}dy\]
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -3966,40 +4167,42 @@
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
-\end_inset
+\end_inset
For
\begin_inset Formula $c=1$
-\end_inset
+\end_inset
this reduces to the normal distribution.
-\layout Section
+\end_layout
+\begin_layout Section
Power-function
-\layout Standard
+\end_layout
+\begin_layout Standard
A special case of the beta distribution with
\begin_inset Formula $b=1$
-\end_inset
+\end_inset
: defined for
\begin_inset Formula $x\in\left[0,1\right]$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
a>0\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;a\right) & = & ax^{a-1}\\
F\left(x;a\right) & = & x^{a}\\
@@ -4010,63 +4213,67 @@
\gamma_{2} & = & \frac{6\left(a^{3}-a^{2}-6a+2\right)}{a\left(a+3\right)\left(a+4\right)}\\
m_{d} & = & 1\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=1-\frac{1}{a}-\log\left(a\right)\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
R-distribution
-\layout Standard
+\end_layout
+\begin_layout Standard
A general-purpose distribution with a variety of shapes controlled by
\begin_inset Formula $c>0.$
-\end_inset
+\end_inset
Range of standard distribution is
\begin_inset Formula $x\in\left[-1,1\right]$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{\left(1-x^{2}\right)^{c/2-1}}{B\left(\frac{1}{2},\frac{c}{2}\right)}\\
F\left(x;c\right) & = & \frac{1}{2}+\frac{x}{B\left(\frac{1}{2},\frac{c}{2}\right)}\,_{2}F_{1}\left(\frac{1}{2},1-\frac{c}{2};\frac{3}{2};x^{2}\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{\left(1+\left(-1\right)^{n}\right)}{2}B\left(\frac{n+1}{2},\frac{c}{2}\right)\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Rayleigh
-\layout Standard
+\end_layout
+\begin_layout Standard
This is Chi distribution with
\begin_inset Formula $L=0.0$
-\end_inset
+\end_inset
and
\begin_inset Formula $\nu=2$
-\end_inset
+\end_inset
and
\begin_inset Formula $S=S$
-\end_inset
+\end_inset
(no location parameter is generally used), the mode of the distribution
is
\begin_inset Formula $S.$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4074,7 +4281,7 @@
F\left(r\right) & = & 1-e^{-r^{2}/2}I_{[0,\infty)}\left(x\right)\\
G\left(q\right) & = & \sqrt{-2\log\left(1-q\right)}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4085,69 +4292,73 @@
m_{d} & = & 1\\
m_{n} & = & \sqrt{2\log\left(2\right)}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=\frac{\gamma}{2}+\log\left(\frac{e}{\sqrt{2}}\right).\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(\frac{n}{2}+1\right)\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Rice*
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined for
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
and
\begin_inset Formula $b>0$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;b\right) & = & x\exp\left(-\frac{x^{2}+b^{2}}{2}\right)I_{0}\left(xb\right)\\
F\left(x;b\right) & = & \int_{0}^{x}\alpha\exp\left(-\frac{\alpha^{2}+b^{2}}{2}\right)I_{0}\left(\alpha b\right)d\alpha\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(1+\frac{n}{2}\right)\,_{1}F_{1}\left(-\frac{n}{2};1;-\frac{b^{2}}{2}\right)\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Reciprocal
-\layout Standard
+\end_layout
+\begin_layout Standard
Shape parameters
\begin_inset Formula $a,b>0$
-\end_inset
+\end_inset
\begin_inset Formula $x\in\left[a,b\right]$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4155,7 +4366,7 @@
F\left(x;a,b\right) & = & \frac{\log\left(x/a\right)}{\log\left(b/a\right)}\\
G\left(q;a,b\right) & = & a\exp\left(q\log\left(b/a\right)\right)=a\left(\frac{b}{a}\right)^{q}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4167,58 +4378,62 @@
m_{d} & = & a\\
m_{n} & = & \sqrt{ab}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=\frac{1}{2}\log\left(ab\right)+\log\left[\log\left(\frac{b}{a}\right)\right].\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Reciprocal Inverse Gaussian
-\layout Standard
+\end_layout
+\begin_layout Standard
The pdf is found from the inverse gaussian (IG),
\begin_inset Formula $f_{RIG}\left(x;\mu\right)=\frac{1}{x^{2}}f_{IG}\left(\frac{1}{x};\mu\right)$
-\end_inset
+\end_inset
defined for
\begin_inset Formula $x\geq0$
-\end_inset
+\end_inset
as
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f_{IG}\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2x\mu^{2}}\right).\\
F_{IG}\left(x;\mu\right) & = & \Phi\left(\frac{1}{\sqrt{x}}\frac{x-\mu}{\mu}\right)+\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{x+\mu}{\mu}\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f_{RIG}\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x}}\exp\left(-\frac{\left(1-\mu x\right)^{2}}{2x\mu^{2}}\right)\\
F_{RIG}\left(x;\mu\right) & = & 1-F_{IG}\left(\frac{1}{x},\mu\right)\\
& = & 1-\Phi\left(\frac{1}{\sqrt{x}}\frac{1-\mu x}{\mu}\right)-\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{1+\mu x}{\mu}\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Semicircular
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined on
\begin_inset Formula $x\in\left[-1,1\right]$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4226,7 +4441,7 @@
F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\left[x\sqrt{1-x^{2}}+\arcsin x\right]\\
G\left(q\right) & = & F^{-1}\left(q\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4235,39 +4450,42 @@
\gamma_{1} & = & 0\\
\gamma_{2} & = & -1\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=0.64472988584940017414.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Studentized Range*
-\layout Section
+\end_layout
+\begin_layout Section
Student t
-\layout Standard
+\end_layout
+\begin_layout Standard
Shape parameter
\begin_inset Formula $\nu>0.$
-\end_inset
+\end_inset
\begin_inset Formula $I\left(a,b,x\right)$
-\end_inset
+\end_inset
is the incomplete beta integral and
\begin_inset Formula $I^{-1}\left(a,b,I\left(a,b,x\right)\right)=x$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\nu\right) & = & \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\pi\nu}\Gamma\left(\frac{\nu}{2}\right)\left[1+\frac{x^{2}}{\nu}\right]^{\frac{\nu+1}{2}}}\\
F\left(x;\nu\right) & = & \left\{ \begin{array}{ccc}
@@ -4277,50 +4495,52 @@
-\sqrt{\frac{\nu}{I^{-1}\left(\frac{\nu}{2},\frac{1}{2},2q\right)}-\nu} & & q\leq\frac{1}{2}\\
\sqrt{\frac{\nu}{I^{-1}\left(\frac{\nu}{2},\frac{1}{2},2-2q\right)}-\nu} & & q\geq\frac{1}{2}\end{array}\right.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
m_{n}=m_{d}=\mu & = & 0\\
\mu_{2} & = & \frac{\nu}{\nu-2}\quad\nu>2\\
\gamma_{1} & = & 0\quad\nu>3\\
\gamma_{2} & = & \frac{6}{\nu-4}\quad\nu>4\end{eqnarray*}
-\end_inset
+\end_inset
As
\begin_inset Formula $\nu\rightarrow\infty,$
-\end_inset
+\end_inset
this distribution approaches the standard normal distribution.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\frac{1}{4}\log\left(\frac{\pi c\Gamma^{2}\left(\frac{c}{2}\right)}{\Gamma^{2}\left(\frac{c+1}{2}\right)}\right)-\frac{\left(c+1\right)}{4}\left[\Psi\left(\frac{c}{2}\right)-cZ\left(c\right)+\pi\tan\left(\frac{\pi c}{2}\right)+\gamma+2\log2\right]\]
-\end_inset
+\end_inset
where
\begin_inset Formula \[
Z\left(c\right)=\,_{3}F_{2}\left(1,1,1+\frac{c}{2};\frac{3}{2},2;1\right)=\sum_{k=0}^{\infty}\frac{k!}{k+1}\frac{\Gamma\left(\frac{c}{2}+1+k\right)}{\Gamma\left(\frac{c}{2}+1\right)}\frac{\Gamma\left(\frac{3}{2}\right)}{\Gamma\left(\frac{3}{2}+k\right)}\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Student Z
-\layout Standard
+\end_layout
+\begin_layout Standard
The student Z distriubtion is defined over all space with one shape parameter
\begin_inset Formula $\nu>0$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4330,7 +4550,7 @@
1-Q\left(x;\nu\right) & & x\geq0\end{array}\right.\\
Q\left(x;\nu\right) & = & \frac{\left|x\right|^{1-n}\Gamma\left(\frac{n}{2}\right)\,_{2}F_{1}\left(\frac{n-1}{2},\frac{n}{2};\frac{n+1}{2};-\frac{1}{x^{2}}\right)}{2\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)}\end{eqnarray*}
-\end_inset
+\end_inset
Interesting moments are
\begin_inset Formula \begin{eqnarray*}
@@ -4339,26 +4559,29 @@
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{6}{\nu-5}.\end{eqnarray*}
-\end_inset
+\end_inset
The moment generating function is
\begin_inset Formula \[
\theta\left(t\right)=2\sqrt{\left|\frac{t}{2}\right|^{\nu-1}}\frac{K_{\left(n-1\right)/2}\left(\left|t\right|\right)}{\Gamma\left(\frac{\nu-1}{2}\right)}.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Symmetric Power*
-\layout Section
+\end_layout
+\begin_layout Section
Triangular
-\layout Standard
+\end_layout
+\begin_layout Standard
One shape parameter
\begin_inset Formula $c\in[0,1]$
-\end_inset
+\end_inset
giving the distance to the peak as a percentage of the total extent of
the non-zero portion.
@@ -4366,7 +4589,7 @@
meter is the width of the non-zero portion.
In standard form we have
\begin_inset Formula $x\in\left[0,1\right].$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4380,7 +4603,7 @@
\sqrt{cq} & & q<c\\
1-\sqrt{\left(1-c\right)\left(1-q\right)} & & q\geq c\end{array}\right.\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4389,24 +4612,26 @@
\gamma_{1} & = & \frac{\sqrt{2}\left(2c-1\right)\left(c+1\right)\left(c-2\right)}{5\left(1-c+c^{2}\right)^{3/2}}\\
\gamma_{2} & = & -\frac{3}{5}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
h\left(X\right) & = & \log\left(\frac{1}{2}\sqrt{e}\right)\\
& \approx & -0.19314718055994530942.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Truncated Exponential
-\layout Standard
+\end_layout
+\begin_layout Standard
This is an exponential distribution defined only over a certain region
\begin_inset Formula $0<x<B$
-\end_inset
+\end_inset
.
In standard form this is
@@ -4415,55 +4640,57 @@
F\left(x;B\right) & = & \frac{1-e^{-x}}{1-e^{-B}}\\
G\left(q;B\right) & = & -\log\left(1-q+qe^{-B}\right)\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\mu_{n}^{\prime}=\Gamma\left(1+n\right)-\Gamma\left(1+n,B\right)\]
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\log\left(e^{B}-1\right)+\frac{1+e^{B}\left(B-1\right)}{1-e^{B}}.\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Truncated Normal
-\layout Standard
+\end_layout
+\begin_layout Standard
A normal distribution restricted to lie within a certain range given by
two parameters
\begin_inset Formula $A$
-\end_inset
+\end_inset
and
\begin_inset Formula $B$
-\end_inset
+\end_inset
.
Notice that this
\begin_inset Formula $A$
-\end_inset
+\end_inset
and
\begin_inset Formula $B$
-\end_inset
+\end_inset
correspond to the bounds on
\begin_inset Formula $x$
-\end_inset
+\end_inset
in standard form.
For
\begin_inset Formula $x\in\left[A,B\right]$
-\end_inset
+\end_inset
we get
\begin_inset Formula \begin{eqnarray*}
@@ -4471,35 +4698,36 @@
F\left(x;A,B\right) & = & \frac{\Phi\left(x\right)-\Phi\left(A\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\
G\left(q;A,B\right) & = & \Phi^{-1}\left[q\Phi\left(B\right)+\Phi\left(A\right)\left(1-q\right)\right]\end{eqnarray*}
-\end_inset
+\end_inset
where
\begin_inset Formula \begin{eqnarray*}
\phi\left(x\right) & = & \frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}\\
\Phi\left(x\right) & = & \int_{-\infty}^{x}\phi\left(u\right)du.\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{\phi\left(A\right)-\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\
\mu_{2} & = & 1+\frac{A\phi\left(A\right)-B\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}-\left(\frac{\phi\left(A\right)-\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\right)^{2}\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Tukey-Lambda
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\lambda\right) & = & F^{\prime}\left(x;\lambda\right)=\frac{1}{G^{\prime}\left(F\left(x;\lambda\right);\lambda\right)}=\frac{1}{F^{\lambda-1}\left(x;\lambda\right)+\left[1-F\left(x;\lambda\right)\right]^{\lambda-1}}\\
F\left(x;\lambda\right) & = & G^{-1}\left(x;\lambda\right)\\
G\left(p;\lambda\right) & = & \frac{p^{\lambda}-\left(1-p\right)^{\lambda}}{\lambda}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4511,50 +4739,52 @@
\mu_{4} & = & \frac{3\Gamma\left(\lambda\right)\Gamma\left(\lambda+\frac{1}{2}\right)2^{-2\lambda}}{\lambda^{3}\Gamma\left(2\lambda+\frac{3}{2}\right)}+\frac{2}{\lambda^{4}\left(1+4\lambda\right)}\\
& & -\frac{2\sqrt{3}\Gamma\left(\lambda\right)2^{-6\lambda}3^{3\lambda}\Gamma\left(\lambda+\frac{1}{3}\right)\Gamma\left(\lambda+\frac{2}{3}\right)}{\lambda^{3}\Gamma\left(2\lambda+\frac{3}{2}\right)\Gamma\left(\lambda+\frac{1}{2}\right)}.\end{eqnarray*}
-\end_inset
+\end_inset
Notice that the
\begin_inset Formula $\lim_{\lambda\rightarrow0}G\left(p;\lambda\right)=\log\left(p/\left(1-p\right)\right)$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \int_{0}^{1}\log\left[G^{\prime}\left(p\right)\right]dp\\
& = & \int_{0}^{1}\log\left[p^{\lambda-1}+\left(1-p\right)^{\lambda-1}\right]dp.\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Uniform
-\layout Standard
+\end_layout
+\begin_layout Standard
Standard form
\begin_inset Formula $x\in\left(0,1\right).$
-\end_inset
+\end_inset
In general form, the lower limit is
\begin_inset Formula $L,$
-\end_inset
+\end_inset
the upper limit is
\begin_inset Formula $S+L.$
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & 1\\
F\left(x\right) & = & x\\
G\left(q\right) & = & q\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4563,45 +4793,47 @@
\gamma_{1} & = & 0\\
\gamma_{2} & = & -\frac{6}{5}\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=0\]
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Von Mises
-\layout Standard
+\end_layout
+\begin_layout Standard
Defined for
\begin_inset Formula $x\in\left[-\pi,\pi\right]$
-\end_inset
+\end_inset
with shape parameter
\begin_inset Formula $b>0$
-\end_inset
+\end_inset
.
Note, the PDF and CDF functions are periodic and are always defined over
\begin_inset Formula $x\in\left[-\pi,\pi\right]$
-\end_inset
+\end_inset
regardless of the location parameter.
Thus, if an input beyond this range is given, it is converted to the equivalent
angle in this range.
For values of
\begin_inset Formula $b<100$
-\end_inset
+\end_inset
the PDF and CDF formulas below are used.
Otherwise, a normal approximation with variance
\begin_inset Formula $1/b$
-\end_inset
+\end_inset
is used.
@@ -4610,51 +4842,53 @@
F\left(x;b\right) & = & \frac{1}{2}+\frac{x}{2\pi}+\sum_{k=1}^{\infty}\frac{I_{k}\left(b\right)\sin\left(kx\right)}{I_{0}\left(b\right)\pi k}\\
G\left(q;b\right) & = & F^{-1}\left(x;b\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & 0\\
\mu_{2} & = & \int_{-\pi}^{\pi}x^{2}f\left(x;b\right)dx\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{\int_{-\pi}^{\pi}x^{4}f\left(x;b\right)dx}{\mu_{2}^{2}}-3\end{eqnarray*}
-\end_inset
+\end_inset
This can be used for defining circular variance.
-\layout Section
+\end_layout
+\begin_layout Section
Wald
-\layout Standard
+\end_layout
+\begin_layout Standard
Special case of the Inverse Normal with shape parameter set to
\begin_inset Formula $1.0$
-\end_inset
+\end_inset
.
Defined for
\begin_inset Formula $x>0$
-\end_inset
+\end_inset
.
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-1\right)^{2}}{2x}\right).\\
F\left(x\right) & = & \Phi\left(\frac{x-1}{\sqrt{x}}\right)+\exp\left(2\right)\Phi\left(-\frac{x+1}{\sqrt{x}}\right)\\
G\left(q;\mu\right) & = & F^{-1}\left(q;\mu\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
-
+\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & 1\\
\mu_{2} & = & 1\\
@@ -4662,24 +4896,27 @@
\gamma_{2} & = & 15\\
m_{d} & = & \frac{1}{2}\left(\sqrt{13}-3\right)\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Wishart*
-\layout Section
+\end_layout
+\begin_layout Section
Wrapped Cauchy
-\layout Standard
+\end_layout
+\begin_layout Standard
For
\begin_inset Formula $x\in\left[0,2\pi\right]$
-\end_inset
+\end_inset
\begin_inset Formula $c\in\left(0,1\right)$
-\end_inset
+\end_inset
\begin_inset Formula \begin{eqnarray*}
@@ -4693,19 +4930,22 @@
r_{c}\left(q\right) & & 0\leq q<\frac{1}{2}\\
2\pi-r_{c}\left(1-q\right) & & \frac{1}{2}\leq q\leq1\end{array}\right.\end{eqnarray*}
-\end_inset
+\end_inset
\begin_inset Formula \[
\]
-\end_inset
+\end_inset
\begin_inset Formula \[
h\left[X\right]=\log\left(2\pi\left(1-c^{2}\right)\right).\]
-\end_inset
+\end_inset
-\the_end
+\end_layout
+
+\end_body
+\end_document
Modified: trunk/scipy/stats/distributions.py
===================================================================
--- trunk/scipy/stats/distributions.py 2008-10-03 14:31:41 UTC (rev 4765)
+++ trunk/scipy/stats/distributions.py 2008-10-03 18:57:20 UTC (rev 4766)
@@ -2280,7 +2280,6 @@
## Log-Laplace (Log Double Exponential)
##
-
class loglaplace_gen(rv_continuous):
def _pdf(self, x, c):
cd2 = c/2.0
@@ -2336,6 +2335,10 @@
lognorm.pdf(x,s) = 1/(s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2)
for x > 0, s > 0.
+
+If log x is normally distributed with mean mu and variance sigma**2,
+then x is log-normally distributed with shape paramter sigma and scale
+parameter exp(mu).
"""
)
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