PropertiesTableGenBetaPrime.tex

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\begin{table*}[p]
\caption[Generalized beta prime distribution -- Properties]{Properties of the generalized beta prime distribution}
\begin{align*}
 \text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad & \text{GenBetaPrime}(x\given a, s, \alpha,\gamma,\beta)   \checked
\\
\text{PDF}\quad &    \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 } \checked
\hspace{-30em}
\\
\text{CDF / CCDF} \quad  &  
\frac{B\big(\alpha, \gamma; (1+(\tfrac{x-a}{s})^{-\beta})^{-1} \big) }{B(\alpha,\gamma)} \checked
\hspace{-7em}
& \tfrac{\beta}{s} >0 \,\big/ \, \tfrac{\beta}{s} <0
\\ 
& \quad = I\Left(  \alpha,\gamma; (1+(\tfrac{x-a}{s})^{-\beta})^{-1} \Right)  \checked
% See McDonald1984, table 3.1 GB2. Expresses incomplete beta function as hypergeometric function.
\\
\text{parameters}\quad &   a,\ s,\ \alpha,\ \gamma,\ \beta \text{ in } \Real \checked 
\\ & \alpha>0, \gamma>0	\checked
\\
\text{support} \quad &    x \geq a &  s > 0 										
\\
&  x\leq a  &  s < 0 
\\
%\text{median} \quad  &  \cdots
%\\
% \text{mode} \quad  & \cdots
% \\
\text{mean} \quad  &   a+\frac{s B(\alpha+\tfrac{1}{\beta},\gamma- \tfrac{1}{\beta}) }{B(\alpha,\gamma)} 
\checked &   -\alpha< \tfrac{1}{\beta} <\gamma \checked
% % See McDonald1984, table 3.1 GB2
\\
\text{variance} \quad  &s^2\Left[\frac{ B(\alpha+\tfrac{2}{\beta},\gamma- \tfrac{2}{\beta}) }{B(\alpha,\gamma)} -  \Left(\frac{ B(\alpha+\tfrac{1}{\beta},\gamma- \tfrac{1}{\beta}) }{B(\alpha,\gamma)}\Right)^2 \Right]  \hspace{-15em}
& -\alpha< \tfrac{2}{\beta} <\gamma \checked
\\
\text{skew} \quad  &  \text{not simple}
\\
\text{ex. kurtosis} \quad  &  \text{not simple}
%\\
%\text{entropy} \quad  &   \ln \frac{1}{B(\alpha, \gamma)} \Left|\frac{\beta}{s}\Right|
%  +(\tfrac{1}{\beta}-\alpha) \big[ \psi(\alpha) - \psi(\gamma)\big]
%  \hspace{-7em} 
%  \\
%& \quad +(\alpha+\gamma) \big[ \psi(\alpha+\gamma) - \psi(\gamma)\big] \checked & \text{\cite[Eq.~(15)]{Tahmasebi2010}} % Not correct?
\\
% \text{MGF} \quad  &  \cdots
% \\
% \text{CF} \quad  &  \cdots
% \\
E[ X^h]  \quad & \frac{|s|^h B(\alpha+\tfrac{h}{\beta},\gamma- \tfrac{h}{\beta}) }{B(\alpha,\gamma)} 
& \!\!a=0,\  -\alpha< \tfrac{h}{\beta} <\gamma \quad   \text{\cite{McDonald1984}} \checked
\end{align*}
\end{table*}