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ChBetaLogistic.tex
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% !TEX encoding = UTF-8 Unicode
% !TEX root = FieldGuide.tex
\Sec{Beta-Logistic Distribution}
\label{sec:BetaLogistic}
\phantomsection
\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Beta-Logistic}
\phantomsection
\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Standard Beta-Logistic}
The {\bf beta-logistic } (Prentice, beta-prime exponential, generalized logistic type IV, exponential generalized beta prime, exponential generalized beta type II, log-F, generalized F, Fisher-z, generalized Gompertz-Verhulst type II) distribution~\cite{Prentice1976, McDonald1991, Johnson1995, Morton2000}
is a four parameter, continuous, univariate, unimodal probability density, with infinite support. The functional form in the most straightforward parameterization is
\begin{align}
\notag
\label{BetaLogistic}
\opr{BetaLogistic}(x\given \pLoc, \pScale,\alpha,\gamma)
& =
\frac{1}{B(\alpha, \gamma) \Left| \pScale\Right|}
\frac{e^{-\alpha \frac{x-\pLoc}{\pScale} }} { \Left(1 + e^{-\frac{x-\pLoc}{\pScale} }\Right)^{\alpha+\gamma} } \checked
\\ &
\ x, \pLoc, \pScale,\alpha,\gamma \text{ in } {\mathbb R}
\\ \notag & \alpha,\gamma >0
\end{align}
The four real parameters consist of a location parameter $\pLoc$, a scale parameter $\pScale$, and two positive shape parameters $\alpha$ and $\gamma$. The {\bf standard beta-logistic} distribution has zero location $\pLoc=0$ and unit scale $\pScale=1$.
The beta-logistic distribution is perhaps most commonly referred to as `generalized logistic', but this terminology is ambiguous, since many types of generalized logistic distribution have been investigated, and this distribution is not `generalized' in the same sense used elsewhere in this survey (See `generalized' \S \ref{sec:notation}). Therefore, we select the name `beta-logistic' as a less ambiguous terminology that mirrors the names beta, beta-prime, and beta-exponential.
\SSec{Special cases}
\dist{Burr type II} (generalized logistic type I, exponential-Burr, skew-logistic) distribution~\cite{Burr1942,Johnson1994}:
\begin{align}
\label{BurrII}
\opr{BurrII}(x\given \pLoc, \pScale, \gamma)
& = \frac{\gamma}{|\pScale|} \frac{e^{- \frac{x-\pLoc}{\pScale} } } { \Left(1 + e^{-\frac{x-\pLoc}{\pScale} }\Right)^{\gamma+1} }
\checked
\\
& = \opr{BetaLogistic}(x\given \pLoc, \pScale, 1, \gamma) \checked
\notag
\end{align}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\textwidth]{pdfBurrII}
\end{center}
\caption[Burr II distributions]{Burr type II distributions, $\opr{BurrII}(x\given0,1,\gamma)$}
\end{figure}
\dist{Reversed Burr type II} (generalized logistic type II) distribution~\cite{Johnson1994}:
\begin{align}
\label{RevBurrII}
\opr{RevBurrII}(x\given \alpha)
& = \frac{\gamma}{|\pScale|} \frac{e^{+ \frac{x-\pLoc}{\pScale} } } { \Left(1 + e^{+\frac{x-\pLoc}{\pScale} }\Right)^{\gamma+1} }
\checked
\\
& = \opr{BurrII}(x\given \pLoc, -\pScale, \gamma) \notag \checked \\
& = \opr{BetaLogistic}(x\given \pLoc, -\pScale, 1, \gamma) \notag \checked \\
& = \opr{BetaLogistic}(x\given \pLoc, +\pScale, \gamma, 1) \notag \checked
\notag
\end{align}
By setting the $\lambda$ parameter to $1$ (instead of $\alpha$) we get a reversed Burr type II.
\begin{table*}[ptb]
\begin{center}
\caption[Beta-logistic distribution -- Special cases]{Special cases of the beta-logistic distribution}
~\\
{\renewcommand{\arraystretch}{1.25}
\begin{tabular}{llccccl}
\eqref{BetaLogistic} & Beta-Logistic & $\pLoc$ & $\pScale$ & $\alpha$ & $\gamma$ \\
\hline
\eqref{BurrII} & Burr type II &. & . & 1 & . & \\
\eqref{RevBurrII}& Reversed Burr type II & . & . & . & 1 &\\
\eqref{CentralLogistic}& Central-Logistic & . & . & $\alpha$ & $\alpha$ & \\
\eqref{Logistic}& Logistic & . & . & 1 & 1 & \\
\eqref{HyperbolicSecant}& Hyperbolic secant & . & . & $\tfrac{1}{2}$ & $\tfrac{1}{2}$ & \\
\end{tabular}
}
\end{center}
\end{table*}
\input{PropertiesTableBetaLogistic}
\pagebreak[4]
\dist{Central-logistic} (generalized logistic type III, symmetric Prentice, symmetric beta-logistic) distribution~\cite{Johnson1995}:
\begin{align}
\label{CentralLogistic}
\opr{CentralLogistic}(x\given \pLoc,\pScale,\alpha)
& =
\frac{1}{B(\alpha, \alpha) |\pScale|}
\frac{e^{-\alpha \frac{x-\pLoc}{\pScale} }} { \Left(1 + e^{-\frac{x-\pLoc}{\pScale} }\Right)^{2\alpha} } \checked
\\ \notag & = \frac{1}{B(\alpha, \alpha) |\pScale|} \bigl[\tfrac{1}{2} \op{sech} \Left(\tfrac{x-\pLoc}{2\pScale} \Right)\bigr]^{2\alpha}
\checked
\\ \notag & = \opr{BetaLogistic}(x\given \pLoc,\pScale,\alpha,\alpha) \checked
\end{align}
With equal shape parameters the beta-logistic is symmetric. This distribution limits to the Laplace distribution~\eqref{Laplace}.
\dist{Logistic} (sech-square, hyperbolic secant square, logit) distribution~\cite{Verhulst1845, Balakrishnan1991, Johnson1995}:
\begin{align}
\label{Logistic}
\opr{Logistic}(x\given \pLoc,\pScale)
& =
\frac{1}{ \Left| \pScale\Right|}
\frac{e^{-\frac{x-\pLoc}{\pScale} }} { \Left(1 + e^{-\frac{x-\pLoc}{\pScale} }\Right)^{2} } \checked
\\ \notag & = \frac{1}{4 |\pScale|} \op{sech}^2 \Left(\frac{x-\pLoc}{\pScale} \Right) \checked
\\ \notag & = \opr{BetaLogistic}(x\given \pLoc,\pScale,1,1) \checked
\end{align}
\dist{Hyperbolic secant} (inverse hyperbolic cosine, inverse cosh) distribution~\cite{Johnson1995,Perks1932,Talacko1956}:
\begin{align}
\label{HyperbolicSecant}
\opr{HyperbolicSecant}(x\given \pLoc, \pScale)
& =
\frac{1}{\pi |\pScale|}
\frac{1}{e^{+\sfrac{x-\pLoc}{2\pScale}} + e^{- \sfrac{x-\pLoc}{2\pScale} }} \checked
\\ & = \frac{1}{2 \pi |\pScale|} \op{sech}(\sfrac{x-\pLoc}{2\lambda}) \notag \checked
\\ \notag & = \opr{BetaLogistic}(x\given \pLoc, \pScale,\tfrac{1}{2},\tfrac{1}{2}) \checked
\end{align}
The hyperbolic secant cumulative distribution function features the Gudermannian sigmoidal function, $\op{gd}(z)$ . \index{Gudermannian function}
\begin{align*}
\op{HyperbolicSecantCDF}(x\given \pLoc, \pScale) & = \frac{1}{\pi}\op{gd}(\frac{x-\pLoc}{2 \pScale}) \checked \\
& = \frac{2}{\pi} \arctan(e^\frac{x-\pLoc}{2\pScale}) - \frac{1}{2} \checked
\end{align*}
The standardized hyperbolic secant distribution (zero mean, unit variance) is $\opr{HyperbolicSecant}(x\given 0, 1/\pi)\checked$.
\newcommand{\oo}{\infty}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\textwidth]{pdfSymBetaLogistic}
\end{center}
\caption[Central-logistic distributions]{Special cases of the symmetric central-logistic distribution~\eqref{CentralLogistic}: Standardized (zero mean, unit variance) normal~($\alpha\rightarrow\oo$), logistic~($\alpha=1$), hyperbolic secant~($\alpha=\tfrac{1}{2}$), and Laplace~($\alpha\rightarrow 0$) (low to high peaks).}
\end{figure}
\SSec{Interrelations}
The beta-logistic distribution arises as a limit of the generalized beta-prime distribution \secref{sec:BetaPrime}. The analogous limit of the generalized beta distribution leads to the beta-exponential family \secref{sec:BetaExp}.
The beta-logistic distribution is the log transform of the beta prime distribution.
\[
\opr{BetaLogistic}(0,1,\alpha,\gamma) \sim - \ln \opr{BetaPrime}(0,1,\alpha,\gamma) \checked
\notag
\]
It follows that beta-logistic variates are related to ratios of gamma variates.
\[
\opr{BetaLogistic}(\pLoc,\pScale,\alpha,\gamma) \sim \pLoc - \pScale \ln \frac{\opr{StdGamma}_1(\gamma)}{\opr{StdGamma}_2(\alpha) }
\notag
\checked
\]
Negating the scale parameter is equivalent to interchanging the two shape parameters.
\[
\opr{BetaLogistic}(x\given \pLoc,+\pScale,\alpha,\gamma) = \opr{BetaLogistic}(x\given \pLoc, - \pScale,\gamma,\alpha) \checked
\notag
\]
The beta-logistic distribution, with integer $\alpha$ and $\gamma$ is the logistic order statistics distribution~\cite{Birnbaum1963,Jones2004}~\secref{OrderStatistic}.
\[
\opr{OrderStatistic}_{\opr{Logistic}(\pLoc,\pScale)} (x \given \gamma, \alpha ) = \opr{BetaLogistic}(x\given \pLoc, \pScale, \alpha, \gamma) \checked
\notag
\]
The beta-logistic limits to the gamma exponential~\eqref{GammaExp} and Laplace \eqref{Laplace} distributions.
\[
\opr{GammaExp}(x\given \nu, \lambda, \alpha) & =
{\lim_{\gamma\rightarrow\infty} \opr{BetaLogistic}(x \given \nu+\lambda/\ln\gamma,\lambda, \alpha, \gamma) }
\checked
\notag
\\
\opr{Laplace}(x\given \eta,\theta) & =
\lim_{\alpha\rightarrow 0} \opr{BetaLogistic}( x\given \eta, \theta\alpha\,\alpha,\alpha) \checked
\notag
\]