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ChGamma.tex
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% !TEX encoding = UTF-8 Unicode
% !TEX root = FieldGuide.tex
\Sec{Gamma Distribution}
\label{sec:Gamma}
\dist{Gamma} ($\Gamma$, Pearson type III) distribution~\cite{Pearson1893, Pearson1895, Johnson1994} :
%
\begin{align}
\label{Gamma}
\opr{Gamma}(x \given a, \theta, \alpha)
&= \frac{1}{\Gamma(\alpha)|\theta|} \Left(\frac{x-a}{\theta}\Right)^{\alpha-1} \exp\Left\{-\frac{x-a}{\theta}\Right\} \checked
\\
\text{for } & x,\ a, \theta, \alpha \In \mathbb{R}, \quad \alpha>0
\notag \checked
\\&= \opr{Amoroso}(x\given a, \theta, \alpha, 1) \notag \checked
\end{align}
The name of this distribution derives from the normalization constant.
\SSec{Special cases}
\phantomsection\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Wein}
\phantomsection\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Erlang}
Special cases of the beta prime distribution are listed in table~\ref{AmorosoTable}, under $\beta=1$.
The gamma distribution often appear as a solution to problems in statistical physics. For example, the energy density of a classical ideal gas, or the {\bf Wien} (Vienna) distribution $\op{Wien}(x\given T)=\opr{Gamma}(x\given 0, T,4)$, an approximation to the relative intensity of black body radiation as a function of the frequency. The {\bf Erlang} (m-Erlang) distribution~\cite{Erlang1909} is a gamma distribution with integer $\alpha$, which models the waiting time to observe $\alpha$ events from a Poisson process with rate $1/\theta$ ($\theta>0$). For $\alpha=1$ we obtain an exponential distribution \eqref{Exp}.
\begin{figure}[tp!]
\begin{center}
\includegraphics[width=\textwidth]{pdfGammaPDF}
\end{center}
\caption[Gamma distributions, unit variance]{Gamma distributions, unit variance $\opr{Gamma}(x\given \tfrac{1}{\alpha},\alpha)$}
\end{figure}
\dist{Standard gamma} (standard Amoroso) distribution~\cite{Johnson1994}:
\[
\opr{StdGamma}(x\given \alpha) & = \frac{1}{\Gamma(\alpha)} x^{\alpha-1} e^{-x} \checked
\label{StdGamma}
\\ & = \opr{Gamma}(x\given 0, 1, \alpha)\notag
\]
\dist{Chi-square} ($\chi^2$) distribution~\cite{Fisher1924,Johnson1994}:
\begin{align}
\label{ChiSqr}
\opr{ChiSqr}(x \given k)
&= \frac{1}{2\Gamma(\tfrac{k}{2})} \Left(\frac{x}{2}\Right)^{\tfrac{k}{2}-1}
\exp\Left\{-\Left(\frac{x}{2}\Right)\Right\} \checked
\\
& \qquad \text{for positive integer } k \notag \checked \\
&= \opr{Gamma}(x\given 0, 2,\tfrac{k}{2}) \notag \checked \\
&= \opr{Stacy}(x\given 2, \tfrac{k}{2},1) \notag \checked \\
&= \opr{Amoroso}(x\given 0, 2, \tfrac{k}{2}, 1) \checked \notag
\end{align}
The distribution of a sum of squares of $k$ independent standard normal random variables. The chi-square distribution is important for statistical hypothesis testing in the frequentist approach to statistical inference.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\textwidth]{pdfChiSqr}
\end{center}
\caption[Chi-square distributions]{Chi-square distributions, $\opr{ChiSqr}(x\given k)$}
\end{figure}
\dist{Scaled chi-square} distribution~\cite{Lee2012}:
\begin{align}
\label{ScaledChiSqr}
\opr{ScaledChiSqr}(x \given \sigma, k)
&= \frac{1}{2\sigma^2\Gamma(\tfrac{k}{2})} \Left(\frac{x}{2\sigma^2}\Right)^{\tfrac{k}{2}-1}
\exp\Left\{-\Left(\frac{x}{2\sigma^2} \Right)\Right\} \checked \\
& \qquad \text{for positive integer } k \notag \\
&= \opr{Stacy}(x\given 2\sigma^2, \tfrac{k}{2},1) \checked\notag \\
&=\opr{Gamma}(x\given 0, 2\sigma^2, \tfrac{k}{2}) \checked \notag \\
&= \opr{Amoroso}(x\given 0, 2\sigma^2, \tfrac{k}{2}, 1) \checked \notag
\end{align}
The distribution of a sum of squares of $k$ independent normal random variables with variance $\sigma^2$.
\begin{table}[tp]
\begin{center}
\caption[Gamma distribution -- Special cases]{Special cases of the gamma family}
\label{GammaTable}
~\\
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{llccccl}
\eqref{Gamma} & gamma & $a$ & $\theta$ & $\alpha$ \\
\hline
\eqref{Gamma} & Erlang & $0$ & $>\!\!0$ & $n$ \\
\eqref{StdGamma} &standard gamma & 0 & 1 & . \\
\eqref{PorterThomas} & Porter-Thomas & 0 & 2 & $\tfrac{1}{2}$ \\
\eqref{ScaledChiSqr} & scaled chi-square & 0 & . & $\tfrac{1}{2}k$ \\
\eqref{ChiSqr} & chi-square & 0 & 2 & $\tfrac{1}{2}k$ \\
\eqref{Exp} & exponential & . & . & $1$ \\
\eqref{Gamma} & Wien & 0 & . & 4 \\
\\
& $(k,\ n\ \text{positive integers})$
\end{tabular}
}
\end{center}
\end{table}
\dist{Porter-Thomas} distribution~\cite{Porter1956a}:
\begin{align}
\label{PorterThomas}
\opr{PorterThomas}(x \given \sigma)
&= \frac{1}{2\sigma^2\Gamma(\tfrac{1}{2})} \Left(\frac{x}{2\sigma^2}\Right)^{-\tfrac{1}{2}}
\exp\Left\{-\Left(\frac{x}{2\sigma^2} \Right)\Right\} \checked \\
&= \opr{Stacy}(x\given 2\sigma^2, \tfrac{1}{2},1) \checked\notag \\
&=\opr{Gamma}(x\given 0, 2\sigma^2, \tfrac{1}{2}) \checked \notag \\
&= \opr{Amoroso}(x\given 0, 2\sigma^2, \tfrac{1}{2}, 1) \checked \notag
\end{align}
A chi-square distribution with a single degree of freedom.
Used to model fluctuations in decay mode strengths of excited nuclei~\cite{Porter1956a}.
\input{PropertiesTableGamma}
\SSec{Interrelations}
\label{GammaInterrelations}
Gamma distributions with common scale obey an addition property:
\begin{align*}
\opr{Gamma}_1(0, \theta, \alpha_1) + \opr{Gamma}_2(0, \theta,\alpha_2) \sim \opr{Gamma}_3(0, \theta,\alpha_1+\alpha_2)
\checked
\end{align*}
The sum of two independent, gamma distributed random variables (with common $\theta$'s, but possibly different $\alpha$'s) is again a gamma random variable~\cite{Johnson1994}.
The Amoroso distribution can be obtained from the standard gamma distribution by the Weibull change of variables, $x \to \Left(\tfrac{x-a}{\theta}\Right)^\beta$.
\[
\opr{Amoroso}(a ,\theta,\alpha,\beta) \sim
a+\theta \Big[{\opr{StdGamma}}(\alpha)\Big]^{1/\beta}
\checked
\notag
\]
For large $\alpha$ the gamma distribution limits to normal~\eqref{Normal}.
\[
\opr{Normal}(x\given \mu,\sigma) =
\lim_{\alpha\rightarrow\infty} \opr{Gamma} (x\given \mu- \sigma\sqrt{\alpha}, \tfrac{\sigma}{\sqrt{\alpha}}, \alpha)
\checked % Checked with MM
\notag
\]
Conversely, the sum of squares of normal distributions is a gamma distribution. See chi-square~\eqref{ChiSqr}.
\begin{align*}
\sum_{i=1,k}\oprr{StdNormal}{Normal}_i()^{2} & \sim \opr{ChiSqr}(k)
\sim \opr{Gamma}(0, 2,\frac{k}{2})
\checked
\notag
\end{align*}
A large variety of distributions can be obtained from transformations of 1~or~2 gamma distributions, which is convenient for generating pseudo-random numbers from those distributions (See appendix~\secref{sec:random}). \label{gammatransforms}
\begin{align*}
\opr{Normal}(\mu,\sigma) &\sim \mu+ \sigma\ \op{Sgn}()\ \sqrt{ 2 \opr{StdGamma}(\half) } & \eqref{Normal}
\checked
\\
\opr{GammaExp}(a,s,\alpha) &\sim a - s \ln\bigl(\opr{StdGamma}(\alpha)\bigr) & \eqref{GammaExp} \checked
\\
\opr{PearsonVII}(a,s, m) & \sim a + s\ \op{Sgn}() \sqrt{ \frac{ \opr{StdGamma}_1(\half)}{\opr{StdGamma}_2(m-\half) } } \hspace{-1.5em}& \eqref{PearsonVII}
\checked
% Follows since HalfPearsonVII is special case of GenBetaPrime
\\
\opr{Cauchy}(a,s) & \sim a + s\ \op{Sgn}() \sqrt{\frac{\opr{StdGamma}_1(\half)}{\opr{StdGamma}_2(\half) } } & \eqref{Cauchy} \checked % Devroye1986 p445
\\
\opr{UnitGamma}(a,s,\alpha,\beta) &\sim a+ s\ \exp \bigl(- \tfrac{1}{\beta}\opr{StdGamma}(\alpha)\bigr) & \eqref{UnitGamma} \checked
\\
\opr{Beta}(a,s,\alpha,\gamma) &
\sim a + s\ \Left(1 + \frac{\opr{StdGamma}_2(\gamma)}{\opr{StdGamma}_1(\alpha)} \Right)^{-1}
& \eqref{Beta} \checked
\\
\opr{BetaPrime}(a,s,\alpha,\gamma) &\sim a + s\ \frac{\opr{StdGamma}_1(\alpha)}{\opr{StdGamma}_2(\gamma) } & \eqref{BetaPrime}
\checked
\\
\opr{Amoroso}(a,\theta,\alpha,\beta)&\sim a + \theta\ \opr{StdGamma}(\alpha)^{\tfrac{1}{\beta}} & \eqref{Amoroso}
\checked
\\
\opr{BetaExp}(a,s, \alpha,\gamma) &
\sim a - s\ \ln \Left(1 + \frac{\opr{StdGamma}_2(\gamma)}{\opr{StdGamma}_1(\alpha)} \Right)^{-1}
& \eqref{BetaExp} \checked
\\
\opr{BetaLogistic}(a,s,\alpha,\gamma) &\sim a - s \ln \Left(\frac{\opr{StdGamma}_1(\alpha)}{\opr{StdGamma}_2(\gamma) } \Right) & \eqref{BetaLogistic} \checked
\\
\opr{GenBeta}(a,s,\alpha,\gamma,\beta) &
\sim a + s \Left(1 + \frac{\opr{StdGamma}_2(\gamma)}{\opr{StdGamma}_1(\alpha)} \Right)^{-\tfrac{1}{\beta}} & \eqref{GenBeta} \checked
\\
\opr{GenBetaPrime}(a,s,\alpha,\gamma,\beta) & \sim a+ s \Left( \frac{\opr{StdGamma}_1(\alpha)}{\opr{StdGamma}_2(\gamma) }\Right)^{\frac{1}{\beta}} & \eqref{GenBetaPrime}
\checked
\end{align*}
Here, $\op{Sgn}()$ is the sign (or Rademacher) discrete random variable: 50\% chance $-1$, 50\% chance $+1$.
\index{Rademacher distribution (discrete)|see{sign distribution}}
\index{sign distribution (discrete)}