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ChBetaPrime.tex
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% !TEX encoding = UTF-8 Unicode
% !TEX root = FieldGuide.tex
\Sec{Beta Prime Distribution}
\label{sec:BetaPrime}
\dist{Beta prime} (beta type II, Pearson type VI, inverse beta, variance ratio, gamma ratio, compound gamma, $\beta'$) distribution~\cite{Pearson1901,Johnson1995}:
\begin{align}
\label{BetaPrime}
\opr{BetaPrime}&(x\given a, s,\alpha, \gamma) \\ \notag&= \frac{1}{B(\alpha,\gamma)}\frac{1}{|s|} \Left(\frac{x-a}{s}\Right)^{\alpha -1} \Left(1+\frac{x-a}{s}\Right)^{-\alpha-\gamma } \checked
\\
&= \opr{GenBetaPrime}(x\given a,s, \alpha,\gamma,1) \notag \checked
\\
& \text{for } a,\ s,\ \alpha,\ \gamma \text{ in } \Real, \ \alpha>0, \gamma>0 \checked
\notag \\
& \text{support } x \geq a \ \text{if}\ s > 0, \ x\leq a \ \text{if}\ s < 0
\notag
\end{align}
A Pearson distribution~\secref{sec:Pearson} with semi-infinite support, and both roots on the real line. Arises notable as the ratio of gamma distributions, and as the order statistics of the uniform-prime distribution~\eqref{UniPrime}.
\SSec{Special cases}
Special cases of the beta prime distribution are listed in table~\ref{GenBetaPrimeTable}, under $\beta=1$.
\dist{Standard beta prime} (beta prime) distribution~\cite{Pearson1901}:
\begin{align}
\label{StdBetaPrime}
\opr{StdBetaPrime}(x\given \alpha, \gamma) &= \frac{1}{B(\alpha,\gamma)} x^{\alpha -1} (1+x )^{-\alpha-\gamma } \checked
\\&= \opr{BetaPrime}(x\given 0,1, \alpha,\gamma) \notag \checked
\\&= \opr{GenBetaPrime}(x\given 0,1, \alpha,\gamma,1) \notag \checked
\end{align}
\begin{figure}[tp!]
\begin{center}
\includegraphics[width=\textwidth]{pdfBetaPrime}
\end{center}
\caption[Beta prime distribution]{A beta prime distribution, $\opr{BetaPrime}(0, 1, 2, 4)$}
\end{figure}
\dist{F} (Snedecor's F, Fisher-Snedecor, Fisher, Fisher-F, variance-ratio, F-ratio) distribution~\cite{Snedecor1934, Aroian1941, Johnson1995}:
\begin{align}
\label{F}
\opr{F}(x\given k_1,k_2) &= \frac{k_1^{\tfrac{k_1}{2} } k_2^{\tfrac{k_2}{2}}}{ B(\tfrac{k_1}{2}, \tfrac{k_2}{2}) }
\frac{x^{\tfrac{k_1}{2} -1}}{(k_2 + k_1 x)^{\tfrac{1}{2}(k_1+k_2)}} \checked
\\ & = \opr{BetaPrime}(x\given 0,\tfrac{k_2}{k_1}, \tfrac{k_1}{2},\tfrac{k_2}{2}) \notag \checked
\\ & = \opr{GenBetaPrime}(x\given 0,\tfrac{k_2}{k_1}, \tfrac{k_1}{2},\tfrac{k_2}{2},1) \checked
\notag
\\ & \text{for positive integers } k_1,\ k_2 \notag \checked
\end{align}
An alternative parameterization of the beta prime distribution that derives from the ratio of two chi-squared distributions \eqref{ChiSqr} with $k_1$ and $k_2$ degrees of freedom.
\[
\opr{F}(k_1,k_2) \sim \frac{\opr{ChiSqr}(k_1) / k_1 }{\opr{ChiSqr}(k_2) / k_2} \checked
\notag
\]
\dist{Inverse Lomax} (inverse Pareto) distribution~\cite{Kleiber2003}:
\begin{align}
\label{InvLomax}
\opr{InvLomax}(x\given a, s,\alpha) &= \frac{\alpha}{|s|} \Left(\frac{x-a}{s}\Right)^{\alpha -1} \Left(1+\frac{x-a}{s}\Right)^{-\alpha-1} \checked
\\ \notag &= \opr{BetaPrime}(x\given a, s, \alpha,1) \checked
\\ \notag &= \opr{GenBetaPrime}(x\given a, s, \alpha,1,1) \checked
\end{align}
\begin{figure}[tp!]
\begin{center}
\includegraphics[width=\textwidth]{pdfInverseLomax}
\end{center}
\caption[Inverse Lomax distribution]{An inverse lomax distribution, $\opr{InvLomax}(0, 1, 2)$}
\end{figure}
\input{PropertiesTableBetaPrime}
\SSec{Interrelations}
The standard beta prime distribution is closed under inversion.
\[
\opr{StdBetaPrime}(\alpha,\gamma) \sim \frac{1}{\opr{StdBetaPrime}(\gamma,\alpha)} \checked
\notag
\]
The beta and beta prime distributions are related by the transformation~\secref{transforms}
\[
\opr{StdBetaPrime}(\alpha,\gamma) \sim \Left( \frac{1}{\opr{StdBeta}(\alpha,\gamma)} -1 \Right)^{-1} \checked
\notag
\]
and, therefore, the generalized beta prime can be realized as a transformation of the standard beta \eqref{StdBeta} distribution.
\[
\opr{GenBetaPrime}(a,s,\alpha,\gamma,\beta) \sim a+ s\Left( \opr{StdBeta}(\alpha,\gamma)^{-1} -1\Right)^{-\sfrac{1}{\beta}}
\checked
\notag
\]
If the scale parameter of a gamma distribution \eqref{Gamma} is also gamma distributed, the resulting compound distribution is beta prime~\cite{Dubey1970}.
\[
\opr{BetaPrime}(0,s,\alpha,\gamma) \sim \opr{Gamma}_2\bigl(0, \opr{Gamma}_1(0, s, \gamma),\alpha \bigr) \checked
\notag
\]
The name {\bf compound gamma} distribution is occasionally used for the anchored beta prime distribution (scale parameter, but no location parameter)
The beta prime distribution is a special case of both the generalized beta~\eqref{GenBeta} and generalized beta prime~\eqref{GenBetaPrime} distributions, and itself limits to the gamma~\eqref{Gamma} and inverse gamma~\eqref{InvGamma} distributions.
\[
\opr{Gamma}(x\given 0, \theta,\alpha)
\notag
& =
\lim_{\gamma\rightarrow\infty} \opr{BetaPrime}(x\given 0, \theta \gamma ,\alpha, \gamma ) \checked
\\
\opr{InvGamma}(x\given \theta,\alpha)
& =
\lim_{\gamma\rightarrow\infty} \opr{BetaPrime}(x\given 0, \theta/ \gamma ,\alpha, \gamma ) \checked
\notag
\]