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ChGenBeta.tex
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% !TEX encoding = UTF-8 Unicode
% !TEX root = FieldGuide.tex
\subpart{Three (or more) shape parameters}
\Sec{Generalized Beta Distribution}
\label{sec:GenBeta}
\phantomsection
\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Generalized beta}
The {\bf generalized beta} (beta-power) distribution~\cite{McDonald1984} is a five parameter, continuous, univariate, unimodal probability density, with finite or semi infinite support. The functional form in the most straightforward parameterizaton is
\begin{align}
\label{GenBeta}
\opr{GenBeta}&(x\given a,s,\alpha, \gamma,\beta)
\\ = &
\frac{1}{B(\alpha, \gamma)} \Left|\frac{\beta}{s}\Right|
\Left(\frac{x-a}{s} \Right)^{\alpha \beta -1} \Left (1-\Left(\frac{x-a}{s}\Right)^\beta\Right)^{\gamma -1}
\notag
\checked
\\
& \text{for } x,\ a,\ \theta,\ \alpha,\ \gamma,\ \beta\ \text{in } \mathbb{R},
\notag
\\ & \alpha>0, \ \gamma >0
\notag
\\ \text{ support } \checked & x \in [a,a+s], s>0,\ \beta>0 \notag
\\ & x\in[a+s,a], s<0,\ \beta>0
\notag
\\ \notag & x\in[a+s,+\infty], s>0,\ \beta<0
\\ \notag & x\in[-\infty,a+s], s<0,\ \beta<0
\notag
\end{align}
The generalized beta distribution arises as the Weibullization of the standard beta distribution, $x\rightarrow (\tfrac{x-a}{s})^{\beta}$, and as the order statistics of the power function distribution~\eqref{PowerFn}. The parameters consist of a location parameter $a$, shape parameter~$s$, Weibull power parameter $\beta$, and two shape parameters~$\alpha$ and~$\gamma$.
\begin{table*}[tp!]
%\addcontentsline{toc}{subsection}{Beta}
\begin{center}
\caption[Generalized beta distributions -- Special cases] {Special cases of generalized beta}
\label{GenBetaTable}
~\\
%Note: Power func special cases go under power function.
{\renewcommand{\arraystretch}{1.25}
\begin{tabular}{llccccc@{\extracolsep{5pt}} l}
\eqref{GenBeta} &generalized beta & $a$ & $s$ & $\alpha$ & $\gamma$ & $\beta$ &
\\ \hline
\eqref{Kumaraswamy} & Kumaraswamy & . & . & 1 & . & . &\\
\\
\eqref{Beta} & beta & . & .& . & . & 1 & \\
\eqref{StdBeta} & standard beta & 0 & 1 & . & . & 1 &\\
\eqref{Beta} & beta, U shaped & . & . & $<\!\!1$ & $<\!\!1$ & 1 &\\
\eqref{Beta} & beta, J shaped & . & . & . & . & 1 & {\small $(\alpha$-$1)(\gamma$-$1) \leq 0$} \\
\eqref{Pert} & pert & a & b-a & $^\dagger$ & $^\dagger$ & 1 & $^\dagger$ See~\eqref{Pert} \\
\eqref{CentralBeta} & central-beta & . & . & $\alpha$ & $\alpha$ & 1 & \\
\eqref{Arcsine} & arcsine & . & . & $\frac{1}{2}$ & $\frac{1}{2}$ & 1 & \\
%\eqref{CentralArcsine}& central arcsine & -$b$ & $2b$ & $\frac{1}{2}$ & $\frac{1}{2}$ & 1 & \\
\eqref{Semicircle}& semicircle & -$b$ & $2b$ & $1\frac{1}{2}$ & $1\frac{1}{2}$ & 1 & \\
\eqref{Epanechnikov}&Epanechnikov & . & . & 2 & 2 & 1 \\
\eqref{Biweight}&biweight & . & . & 3 & 3 & 1 \\
\eqref{Triweight}&triweight & . & . & 4 & 4 & 1 \\
%\eqref{PowerFn} & power function & . & . & . & 1 & 1& \\
%\eqref{PowerFn} & reciprocal & . & . & 0 & 1 & 1& \\
%\eqref{PowerFn} & Pearson type VIII & 0 & . & $<\!\!1$ & 1& 1&\\
%\eqref{PowerFn} & Pearson type IX & 0 & . & $>\!\!1$ & 1 & 1&\\
\eqref{PearsonXII} & Pearson XII & . & . & . & 2-$\alpha$&1& $\alpha<2$ \\
%\eqref{Wedge} & wedge & . & . & 2 & 1 & 1 &\\
\\
\eqref{BetaPrime} & beta-prime & . & . & . & . & -1 \\
\eqref{PowerFn} & power function & . & . & 1 & 1 & . & \\
\eqref{Uniform} & uniform & . & . & 1 & 1 & 1 &\\
\eqref{Uniform} & standard uniform & 0 & 1 & 1 & 1 & 1 &\\
%\\
%& \underline{Limits}
%\\
%\eqref{UnitGamma} & unit gamma & . & . & $\alpha$ &.& $\tfrac{\delta}{\alpha}$ & $\lim_{\alpha\rightarrow\infty }$ \\
%\eqref{Amoroso} & Amoroso & . & $\theta \gamma^{\frac{1}{\beta}}$& . & $\gamma$ & . & $\lim_{\gamma\rightarrow\infty}$ \\
%\eqref{BetaExp} & beta exp. & ${\pLoc\text{-}\beta\pScale}$ & $\beta\pScale$ & . & . &$\beta$ & $\lim_{\beta\rightarrow\infty}$ \\
\end{tabular}
}
\end{center}
\end{table*}
\input{PropertiesTableGenBeta}
\SSec{Special Cases}
The beta distribution ($\beta$=1) and specializations are described in \secref{sec:Beta}.
\dist{Kumaraswamy} (minimax) distribution~\cite{Kumaraswamy1980, Leemis2008, Jones2009}:
\begin{align}
\label{Kumaraswamy}
\opr{Kumaraswamy}(x\given a,s,\gamma,\beta) &= \gamma \Left|\frac{\beta}{s}\Right| \Left(\frac{x-a}{s}\Right)^{\beta-1} \Left(1-\Left(\frac{x-a}{s}\Right)^{\beta}\Right)^{\gamma-1} \checked
\\
&= \opr{GenBeta}(x\given a,s,1,\gamma,\beta) \notag \checked
\end{align}
Proposed as an alternative to the beta distribution for modeling bounded variables, since the cumulative distribution function has a simple closed form,
\[\op{KumaraswamyCDF}(x\given 0, 1, \gamma,\beta) = 1- (1-x^{\beta})^\gamma. \checked \notag\]
\begin{figure}[tp!]
\begin{center}
\includegraphics[width=\textwidth]{pdfKumaraswamy}
\end{center}
\caption[Kumaraswamy distribution]{A Kumaraswamy distribution, $\opr{Kumaraswamy}(0, 1, 2, 4)$}
\end{figure}
\SSec{Interrelations}
The generalized beta distribution describes the order statistics of a power function distribution \eqref{PowerFn}.
\begin{align*}
\opr{OrderStatistic}_{\opr{PowerFn}(a,s,\beta)} &(x \given \alpha, \gamma) = \opr{GenBeta}(x\given a, s, \alpha,\gamma, \beta)
\checked
\end{align*}
Conversely, the power function \eqref{PowerFn} distribution is a special case of the generalized beta distribution.
\begin{align*}
\opr{GenBeta}(x\given a, s, 1, 1,\beta) & = \opr{PowerFn}(x\given a,s,\beta) \checked
\notag
\end{align*}
Setting $\beta=1$ yields the beta distribution \eqref{Beta},
\[
\opr{GenBeta}(x\given a,s,\alpha, \gamma,1) = \opr{Beta}(x\given a,s,\alpha,\gamma) \ , \checked
\notag
\]
and setting $\beta=-1$ yields the beta prime (or inverse beta) distribution \eqref{BetaPrime},
\[
\opr{GenBeta}(x\given a,s,\alpha, \gamma,-1) = \opr{BetaPrime}(x\given a+s,s,\gamma,\alpha) \ . \checked
\notag
\]
The beta \secref{sec:Beta} and beta prime \secref{sec:BetaPrime} distributions have many named special cases, see tables~\ref{GenBetaTable} and \ref{GenBetaPrimeTable}.
The unit gamma distribution~\eqref{UnitGamma} arises in the limit $\lim_{\beta\rightarrow0}$ with $\alpha\beta=\text{constant}$,
\[
\lim_{\beta\rightarrow0 } \opr{GenBeta}(x\given a,s,\tfrac{\delta}{\beta}, \gamma,\beta) = \opr{UnitGamma}(x\given a, s, \gamma,\delta) \ . \checked
\notag
\]
In the limit $\gamma\rightarrow\infty$ (or equivalently $\alpha\rightarrow\infty$) we obtain the Amoroso distribution \eqref{Amoroso} with semi-infinite support, the parent of the gamma distribution family~\cite{McDonald1984},
\[
\lim_{\gamma\rightarrow\infty} \opr{GenBeta}(x\given a, \theta \gamma^{\frac{1}{\beta}} ,\alpha, \gamma, \beta ) = \opr{Amoroso}(x\given a,\theta,\alpha, \beta) \ . \checked
\notag
\]
The limit $\lim_{\beta\rightarrow+\infty }$ yields the beta-exponential distribution~\eqref{BetaExp}
\[
\lim_{\beta\rightarrow+\infty } \opr{GenBeta}(x\given \pLoc+\beta\lambda,-\beta\pScale,\alpha,\gamma,\beta) = \opr{BetaExp}(x\given \pLoc,\pScale,\alpha, \gamma) \checked
\ .
\notag
\]