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ChBetaExp.tex
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% !TEX encoding = UTF-8 Unicode
% !TEX root = FieldGuide.tex
\Sec{Beta-Exponential Distribution}
\label{sec:BetaExp}
\phantomsection
\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Beta-exponential}
\phantomsection
\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Standard beta-exponential}
The {\bf beta-exponential} (Gompertz-Verhulst, generalized Gompertz-Verhulst type III,
log-beta, exponential generalized beta type I) distribution~\cite{Ahuja1967,Nadarajah2006, Iyer-Biswas2014a} is a four parameter, continuous, univariate, unimodal probability density, with semi-infinite support. The functional form in the most straightforward parameterization is
\begin{align}
\label{BetaExp}
\opr{BetaExp}(x\given \pLoc,\pScale,\alpha, \gamma) =&
\frac{1}{B(\alpha, \gamma)} \frac{1}{|\pScale|}\
e^{-\alpha \frac{x-\pLoc}{\pScale} } \Left(1 - e^{-\frac{x-\pLoc}{\pScale} }\Right)^{\gamma-1}
\checked
\\ \notag
& \text{for } x,\ \pLoc,\ \pScale,\ \alpha,\ \gamma \text{ in } \mathbb{R}, \checked
\\ \notag & \alpha,\ \gamma >0,\quad \tfrac{x-\pLoc}{\pScale} >0 \ .\checked
\end{align}
The four real parameters of the beta-exponential distribution consist of a location parameter $\pLoc$, a scale parameter $\pScale$, and two positive shape parameters $\alpha$ and $\gamma$. The {\bf standard beta-exponential} distribution has zero location $\pLoc=0$ and unit scale $\pScale=1$.
\begin{figure}[p]
\begin{center}
\includegraphics[width=\textwidth]{pdfBetaExp}
\end{center}
\caption[Beta-exponential distributions]{Beta-exponential distributions, (a) $\opr{BetaExp}(x\given 0,1,2, 2)$,
(b) $\opr{BetaExp}(x\given 0, 1, 2, 4)$,
(c) $\opr{BetaExp}(x\given 0, 1, 2, 8)$.
}
\end{figure}
\begin{figure}[p]
\begin{center}
\includegraphics[width=\textwidth]{pdfExpExp}
\end{center}
\caption[Exponentiated exponential distribution]{Exponentiated exponential distribution, $\opr{ExpExp}(x\given 0,1,2)$.
}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{pdfSinhNK}
\end{center}
\caption[Hyperbolic sine and Nadarajah-Kotz distributions.]{Hyperbolic sine $\opr{HyperbolicSine}(x\given \tfrac{1}{2})$ and Nadarajah-Kotz $\opr{NadarajahKotz}(x)$ distributions. }
\end{figure}
This distribution has a similar shape to the gamma \eqref{Gamma} distribution. Near the boundary the density scales like $x^{\gamma-1}$, but decays exponentially in the wing.
\SSec{Special cases}
\dist{Exponentiated exponential} (generalized exponential, Verhulst) distribution~\cite{Verhulst1847,Ahuja1967,Gupta2001}:
\begin{align}
\label{ExpExp}
\opr{ExpExp}(x\given \pLoc,\pScale,\gamma)
&= \frac{\gamma}{ \Left|\pScale\Right|}
e^{- \frac{x-\pLoc}{\pScale} } \Left(1 - e^{-\frac{x-\pLoc}{\pScale} }\Right)^{\gamma-1} \checked
\\ & = \opr{BetaExp}(x\given \pLoc,\pScale,1, \gamma) \notag \checked
\end{align}
A special case similar in shape to the gamma or Weibull \eqref{Weibull} distribution. So named because the cumulative distribution function is equal to the exponential distribution function raise to a power.
\[
\op{ExpExpCDF}(x\given \pLoc,\pScale,\gamma) = \big[\op{ExpCDF}(x\given \pLoc,\pScale)\big]^{\gamma}
\checked
\notag
\]
\begin{table*}[bt]
\begin{center}
\caption[Beta-exponential distribution -- Special cases]{Special cases of the beta-exponential family}
~\\
{\renewcommand{\arraystretch}{1.25}
\begin{tabular}{llccccl}
\eqref{BetaExp} & beta-exponential & $\pLoc$ & $\pScale$ & $\alpha$ & $\gamma$ \\
\hline
& std. beta-exponential & $0$ & $1$ & . & . \\
\eqref{ExpExp} & exponentiated exponential & . & . & $1$ & . \\
\eqref{HyperbolicSine} & hyperbolic sine & . & . & $\tfrac{1}{2}(1\text{-}\gamma)$ & $\gamma$ & $0<\gamma<1$ \\
\eqref{NadarajahKotz} & Nadarajah-Kotz & . & . & $\tfrac{1}{2}$ & $\tfrac{1}{2}$ \\
\eqref{Exp} & exponential & . & . & 1 & $1$ \\
\end{tabular} }
\end{center}
\end{table*}
\input{PropertiesTableBetaExp}
\dist{Hyperbolic sine} distribution~\cite{\self}:
\begin{align}
\label{HyperbolicSine}
\opr{HyperbolicSine}(x\given \pLoc, \pScale, \gamma)
&= \frac{1}{ B(\tfrac{1-\gamma}{2}, \gamma)}\frac{1}{|\pScale|} \bigl( e^{+\frac{x-\pLoc}{2 \pScale}} - e^{-\frac{x-\pLoc}{ 2\pScale}} \bigr)^{\gamma-1}
\checked
\\ \notag &= \frac{2^{\gamma-1}}{ B(\frac{1-\gamma}{2}, \gamma) |\pScale| } \bigl(\op{sinh}(\sfrac{x-\pLoc}{2\pScale})\bigr)^{\gamma-1}
\checked
\\ \notag & = \opr{BetaExp}(x\given \pLoc,\pScale ,\tfrac{1-\gamma}{2}, \gamma), \quad 0<\gamma<1
\checked
\end{align}
Compare to the hyperbolic secant distribution \eqref{HyperbolicSecant}.
\dist{Nadarajah-Kotz} distribution~\cite{Nadarajah2006,\self} :
\begin{align}
\label{NadarajahKotz}
\opr{NadarajahKotz}(x\given \pLoc, \pScale)
&= \frac{1}{\pi |\pScale|} \frac{1}{\sqrt{e^{\frac{x-\pLoc}{\pScale}} -1}} \checked
\\ \notag & = \opr{BetaExp}(x\given \pLoc,\pScale, \tfrac{1}{2}, \tfrac{1}{2} ) \checked
\end{align}
A notable special case when $\alpha=\gamma=\tfrac{1}{2}$. The cumulative distribution function has the simple form
\[
\op{NadarajahKotzCDF}(x\given 0, 1)= \frac{2}{\pi} \op{arctan} \sqrt{\exp(x) -1} \, . \checked
\notag
\]
% Self citation is for name of distribution
\SSec{Interrelations}
The beta-exponential distribution is a limit of the generalized beta distribution \secref{sec:Beta}. The analogous limit of the generalized beta prime distribution \secref{sec:BetaPrime} results in the beta-logistic family of distributions~\secref{sec:BetaLogistic}.
The beta-exponential distribution is the log transform of the beta distribution~\eqref{Beta}.
\[
\oprr{StdBetaExp}{BetaExp}(\alpha,\gamma) \sim - \ln\bigl( \opr{StdBeta}(\alpha,\gamma) \bigr)
\checked \notag
\]
It follows that beta-exponential variates are related to ratios of gamma variates.
\[
\oprr{StdBetaExp}{BetaExp}(\alpha,\gamma) \sim - \ln \frac{\opr{StdGamma}_1(\alpha)}{\opr{StdGamma}_1(\alpha)+ \opr{StdGamma}_2(\gamma) }
\checked \notag
\]
The beta-exponential distribution describes the order statistics \secref{OrderStatistic} of the exponential distribution \eqref{Exp}.
\begin{align*}
\opr{OrderStatistic}_{\opr{Exp}(\pLoc,\pScale)} & (x \given \gamma,\alpha) = \opr{BetaExp}(x\given \pLoc, \pScale, \alpha, \gamma) \checked
\end{align*}
With $\gamma=1$ we recover the exponential distribution.
\[
\opr{BetaExp}(x\given \pLoc, \pScale, \alpha,1) = \opr{Exp}(x\given \pLoc, \tfrac{\pScale}{\alpha})
\checked \notag
\]
The beta-exponential distribution is a limit of the generalized beta distribution~\eqref{GenBeta}, and itself limits to the gamma-exponential distriution~\eqref{GammaExp}.
\[
\notag
\opr{GammaExp}(x\given \nu, \lambda, \alpha) & =
{\lim_{\gamma\rightarrow\infty} \opr{BetaExp}(x \given \nu+\lambda/\ln\gamma, \lambda, \alpha, \gamma) }
\checked \notag
\]