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ChGenBetaPrime.tex
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ChGenBetaPrime.tex
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% !TEX encoding = UTF-8 Unicode
% !TEX root = FieldGuide.tex
\Sec[Gen. Beta Prime Distribution] {Generalized Beta Prime Distribution}
\label{sec:GenBetaPrime}
\phantomsection
\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Generalized beta prime}
The {\bf generalized beta-prime} (Feller-Pareto, beta-log-logistic, generalized\linebreak gamma ratio, Majumder-Chakravart, generalized beta type II, generalized Feller-Pareto) distribution~\cite{Feller1971, McDonald1984, Tahmasebi2010} is a five parameter, continuous, univariate, unimodal probability density, with semi-infinite support. The functional form in the most straightforward parameterization is
\begin{align}
\label{GenBetaPrime}
\opr{GenBetaPrime}&(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)^{-\alpha-\gamma }
\notag
\checked
\\ & \quad a,\ s,\ \alpha,\ \gamma,\ \beta \text{ in } \mathbb{R},\quad \alpha,\ \gamma >0
\notag
\end{align}
The five real parameters of the generalized beta prime distribution consist of a location parameter~$a$,
scale parameter~$s$, two shape parameters,~$\alpha$ and~$\gamma$, and the Weibull power parameter $\beta$. The shape parameters, $\alpha$ and $\gamma$, are positive.
The generalized beta prime arises as the Weibull transform of the standard beta prime distribution \eqref{StdBetaPrime}, and as order statistics of the log-logistic distribution. The Amoroso distribution is a limiting form, and a variety of other distributions occur as special cases. (See Table~\ref{GenBetaPrimeTable}). These distributions are most often encountered as parametric models for survival statistics developed by economists and actuaries.
\SSec{Special cases}
\dist {Transformed beta} distribution~\cite{McDonald1984,Klugman2012}:
\begin{align}
\label{TransformedBeta}
\opr{TransformedBeta}&(x\given s, \alpha,\gamma,\beta)
\\ \notag & =
\frac{1}{B(\alpha, \gamma)} \Left|\frac{\beta}{s}\Right|
\Left(\frac{x}{s}\Right)^{\alpha\beta -1} \Left(1+ \Left(\frac{x}{s}\Right)^\beta \Right)^{-\alpha-\gamma }
\checked
\\ \notag &= \opr{GenBetaPrime}(x\given 0, s, \alpha,\gamma,\beta)
\checked
\end{align}
A generalized beta prime distribution without a location parameter, $a=0$.
\dist{Burr} (Burr type XII, Pareto type IV, beta-P, Singh-Maddala, generalized log-logistic, exponential-gamma,Weibull-gamma)
%q-exponential\sscite{Nadarajah2006}) %This dosn't compute.
distribution~\cite{Burr1942,Tadikamalla1980, Kleiber2003}:
\begin{align}
\label{Burr}
\opr{Burr}(x\given a, s, \gamma,\beta)
&=
\frac{\beta \gamma}{|s|} \Left(\frac{x-a}{s}\Right)^{\beta-1} \Left(1+\Left(\frac{x-a}{s}\Right)^\beta\Right)^{-\gamma-1}
\checked
\\ & = \opr{GenBetaPrime}(x\given a, s, 1,\gamma,\beta) \notag \checked
\end{align}
Most commonly encountered as a model of income distribution.
\begin{table*}[tp]
%\addcontentsline{toc}{subsection}{Beta prime}
\begin{center}
\caption[Generalized beta prime distribution -- Special cases]{Special cases of generalized beta prime}
\label{GenBetaPrimeTable}
~\\
{\renewcommand{\arraystretch}{1.25}
\begin{tabular}{llccccc}
\eqref{GenBetaPrime} & generalized beta prime & $a$ & $s$ & $\alpha$ & $\gamma$ & ${\beta}$ \\
\hline
\eqref{Burr} & Burr & . & . & 1 & . & . \\
\eqref{Dagum} & Dagum & . & . & . & 1 & . \\
\eqref{Paralogistic} & paralogistic & . & . & 1 & $\beta$ & . \\
\eqref{InvParalogistic} & inverse paralogistic & . & . & $\beta$& 1 & . \\
\eqref{LogLogistic} & log-logistic & . & . & 1 & 1 & . \\
\eqref{GenBetaPrime} & transformed beta & 0 & . & . & . & . \\
\eqref{HalfGenPearsonVII} & half gen. Pearson VII & . & . & $\tfrac{1}{\beta}$ & $m$-$\tfrac{1}{\beta}$ & . \\
\eqref{BetaPrime} & beta prime & . & . & . & . & 1 \\
\eqref{Lomax} & Lomax & . & . & 1 & . & 1 \\
\eqref{InvLomax} & inverse Lomax & . & . & . & 1 & 1 \\
\eqref{StdBetaPrime} & std.~beta-prime & 0 & 1 & . & . & 1 \\
\eqref{F} & F & 0 & $\tfrac{k_2}{k_1}$ & $\tfrac{k_1}{2}$ & $\tfrac{k_2}{2}$ & 1 \\
\eqref{UniPrime} & uniform-prime & . & . & 1 & 1 & 1 \\
\eqref{ExpRatio} & exponential ratio & 0 & . & 1 & 1 & 1 \\
\eqref{HalfPearsonVII} & half-Pearson VII & . & . & $\tfrac{1}{2}$ & . & 2 \\
\eqref{HalfCauchy} & half-Cauchy & . & . & $\tfrac{1}{2}$ & $\tfrac{1}{2}$ & 2 \\
\end{tabular}
}
\end{center}
\end{table*}
\input{PropertiesTableGenBetaPrime}
\pagebreak
\dist{Dagum} (Inverse Burr, Burr type III, Dagum type I, beta-kappa, beta-k, Mielke) distribution~\cite{Burr1942, Dagum1977, Tadikamalla1980}:
\begin{align}
\label{Dagum}
\opr{Dagum}(x \given a, s, \gamma, \beta)
%&= \gamma \beta \frac{x^{\gamma\beta-1}}{ (1+x^\beta)^{\gamma +1} } \\
&= \frac{\beta \gamma}{|s|} \Left(\frac{x-a}{s}\Right)^{\gamma \beta-1} \Left(1+\Left(\frac{x-a}{s}\Right)^\beta\Right)^{-\gamma-1}
\checked
%\\ \notag &= \opr{Burr}(x\given a,s, \gamma,-\beta) \checked % Burr restricted to positive beta
\\ \notag &= \opr{GenBetaPrime}(x\given a, s, 1, \gamma,-\beta) \checked
\\ \notag &= \opr{GenBetaPrime}(x\given a, s, \gamma,1,+\beta) \checked
\end{align}
\dist{Paralogistic} distribution~\cite{Kleiber2003}:
\begin{align}
\label{Paralogistic}
\opr{Paralogistic}(x\given a, s, \beta) &= \frac{\beta^2}{|s|} \frac{ \Left(\frac{x-a}{s}\Right)^{\beta-1}}{ (1+ \Left(\frac{x-a}{s}\Right)^\beta)^{\beta+1} } \checked
\\ \notag &= \opr{GenBetaPrime}(x\given a, s, 1,\beta,\beta)
\checked
\end{align}
\dist{Inverse paralogistic} distribution~\cite{Klugman2012}:
\begin{align}
\label{InvParalogistic}
\opr{InvParalogistic}(x\given a,s,\beta) &= \frac{\beta^2}{|s|} \frac{ \Left(\frac{x-a}{s}\Right)^{\beta^2-1}}{ (1+ \Left(\frac{x-a}{s}\Right)^\beta)^{\beta+1} } \checked
\\ \notag &= \opr{GenBetaPrime}(x\given a, s, \beta,1,\beta) \checked
\end{align}
\dist{Log-logistic} (Fisk, Weibull-exponential, Pareto type III, power prime) distribution~\cite{Shah1963, Johnson1995, McDonald1995}:
\begin{align}
\label{LogLogistic}
\opr{LogLogistic}(x\given a,s,\beta) &=\Left|\frac{\beta}{s}\Right| \frac{\Left(\frac{x-a}{s}\Right)^{\beta-1}}{ \Left(1+\Left(\frac{x-a}{s}\Right)^{\beta}\Right)^{2} } \checked
\\ \notag &= \opr{Burr}(x\given a, s, 1,\beta) \checked
\\ \notag &= \opr{GenBetaPrime}(x\given a, s, 1,1,\beta) \checked
\end{align}
Used as a parametric model for survival analysis and, in economics, as a model for the distribution of wealth or income.
The logistic and log-logistic distributions are related by an exponential transform.
\[
\opr{LogLogistic}(0,s,\beta) &\sim \exp\bigl(-\opr{Logistic}(-\ln s,\sfrac{1}{\beta})\bigr)
\checked
\notag
\]
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{pdfLogLogistic}
\end{center}
\caption[Log-logistic distributions]{Log-logistic distributions, $\opr{LogLogistic}(x\given 0,1,\beta)$.}
\end{figure}
\dist{Half-Pearson VII} (half-t) distribution~\cite{Gelman2006}:
\begin{align}
\label{HalfPearsonVII}
\opr{HalfPearsonVII}&(x\given a, s, m) \\
\notag &=
\frac{1}{B(\tfrac{1}{2},m-\tfrac{1}{2})} \frac{2}{|s|}
\Left(1+ \Left(\frac{x-a}{s}\Right)^2 \Right)^{-m} \checked
\\ \notag & = \opr{GenBetaPrime}(x\given a, s, \tfrac{1}{2},m-\tfrac{1}{2}, 2) \checked
\end{align}
The Pearson type VII \eqref{PearsonVII} distribution truncated at the center of symmetry. Investigated as a prior for variance parameters in hierarchal models~\cite{Gelman2006}.
\dist{Half-Cauchy} distribution~\cite{Gelman2006}:
\begin{align}
\label{HalfCauchy}
\opr{HalfCauchy}(x\given a, s) &=
\frac{2}{\pi |s|}
\Left(1+ \Left(\frac{x-a}{s}\Right)^2 \Right)^{-1}
\checked
\\ \notag & = \opr{HalfPearsonVII}(x\given a, s, 1) \checked
\\ \notag & = \opr{GenBetaPrime}(x\given a, s, \tfrac{1}{2},\tfrac{1}{2}, 2)
\checked
\end{align}
A notable subclass of the Half-Pearson type VII, the Cauchy distribution \eqref{Cauchy} truncated at the center of symmetry.
\dist{Half generalized Pearson VII} distribution~\cite{\self}:
\begin{align}
\label{HalfGenPearsonVII}
\opr{HalfGenPearsonVII}&(x\given a,s, m,\beta)
\\ \notag = & \frac{\beta}{ |s| B(m-\frac{1}{\beta}, \frac{1}{\beta} )} \Left( 1 +\Left( \frac{x-a}{s}\Right)^{\beta} \Right)^{-m}
\checked
\\ \notag & = \opr{GenBetaPrime}(x\given a, s, \tfrac{1}{\beta},m-\tfrac{1}{\beta}, \beta)
\checked
%\\ \notag & x, a,s, m,\beta \text{ in } {\mathbb R} \\
%& \tfrac{x-a}{s}>0,\ \beta>0,\ m>0,\ \beta m >1
%\notag
\end{align}
One half of a Generalized Pearson VII distribution~\eqref{GenPearsonVII}.
Special cases include half Pearson VII \eqref{HalfPearsonVII}, half Cauchy \eqref{HalfCauchy}, {\bf half-Laha} (See \eqref{Laha}), and uniform prime \eqref{UniPrime} distributions.
\begin{align*}
\opr{HalfGenPearsonVII}(x\given a,s, m,2) &= \opr{HalfPearsonVII}(x\given a,s,m) \checked \\
\opr{HalfGenPearsonVII}(x\given a,s, 1,2) &= \opr{HalfCauchy}(x\given a,s) \checked \\
\opr{HalfGenPearsonVII}(x\given a,s, 1,4) &= \oprr{HalfLaha}{Laha}(x\given a,s) \checked \\
\opr{HalfGenPearsonVII}(x\given a,s, 2,1) &= \opr{UniPrime}(x\given a,s) \checked
\end{align*}
The half exponential power \eqref{HalfExpPower} distribution occurs in the large $m$ limit.
\begin{align*}
\lim_{m\rightarrow\infty} \opr{HalfGenPearsonVII}&(x\given a,\theta m^{\sfrac{1}{\beta}}, m,\beta)
= \opr{HalfExpPower}(x\given a,\theta,\beta) \checked
\end{align*}
\phantomsection
\addcontentsline{toc}{subsection}{~~~~~~~~~~~~Half-Laha}
% ====================================================================
\SSec{Interrelations}
Negating the Weibull parameter of the generalized beta prime distribution is equivalent to exchanging the shape parameters $\alpha$ and $\gamma$.
\begin{align*}
\opr{GenBetaPrime}&(x\given a, s, \alpha,\gamma,\beta) = \opr{GenBetaPrime}(x\given a, s,\gamma, \alpha,-\beta) \checked
\end{align*}
The distribution is related to ratios of gamma distributions.
\[
\opr{GenBetaPrime}(a,s,\alpha,\gamma,\beta) \sim a+ s\Left( \frac{\opr{StdGamma}_1(\alpha)}{\opr{StdGamma}_2(\gamma) } \Right)^{\tfrac{1}{\beta}} \checked
\notag
\]
Limit of the generalized beta prime distribution include the Amoroso \eqref{Amoroso}~\cite{McDonald1984} and beta-logistic \eqref{BetaLogistic} distributions.
\begin{align*}
\lim_{\gamma\rightarrow\infty} \opr{GenBetaPrime}(x\given a, \theta \gamma^{\frac{1}{\beta}} ,\alpha, \gamma, \beta )
%\\
& = \opr{Amoroso}(x\given a,\theta,\alpha, \beta) \checked \\
\lim_{\beta\rightarrow\infty} \opr{GenBetaPrime}(x\given \pLoc+\beta\pScale, -\beta \pScale, \alpha, \gamma, \beta)
%\\
& = \opr{BetaLogistic}(x\given \pLoc,\pScale, \gamma, \alpha) \checked
\end{align*}
Therefore, the generalized beta prime also indirectly limits to the normal \eqref{Normal}, log-normal \eqref{LogNormal}, gamma-exponential \eqref{GammaExp}, Laplace \eqref{Laplace} and power-function \eqref{PowerFn} distributions, among others.
Generalized beta prime describes the order statistics \secref{OrderStatistic} of the log-logistic distribution \eqref{LogLogistic}).
\begin{align*}
\opr{OrderStatistic}_{\opr{LogLogistic}(a,s,\beta)}(x \given \gamma, \alpha ) & = \opr{GenBetaPrime}(x\given a, s, \alpha, \gamma, \beta) \checked
\end{align*}
Despite occasional claims to the contrary,
the log-Cauchy distribution is not a special case of the generalized beta prime distribution (generalized beta prime is mono-modal, log-Cauchy is not).
% =================================================================================