PropertiesTableBetaPrime.tex

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\begin{table*}[tp]
\caption[Beta prime distribution -- Properties] {Properties of the beta prime distribution}
\begin{align*}
 \text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad & \op{BetaPrime}(x\given a, s, \alpha,\gamma)  	\checked
\\
\text{PDF}\quad &    \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
\hspace{-2em}
\\
\text{CDF / CCDF} \quad  &  
\frac{B\big(\alpha, \gamma; (1+(\tfrac{x-a}{s})^{-1})^{-1} \big) }{B(\alpha,\gamma)} \checked
\hspace{-8em}
& s >0 \,\big/ \, s<0
\\ 
& \quad = I\Left(  \alpha,\gamma; (1+(\tfrac{x-a}{s})^{-1})^{-1} \Right) 			\checked
\\
\text{parameters}\quad &   a,\ s,\ \alpha,\ \gamma, \text{ in } \Real \checked 
\\ & \alpha>0, \gamma>0 \checked
\\
\text{support} \quad &    x \geq a &  s > 0 			\checked
\\
&  x\leq a  &  s < 0 							\checked
\\
%\text{median} \quad  &  \cdots
%\\
\text{mode} \quad  & a + s \frac{\alpha-1}{\gamma+1} & \alpha\geq 1 \checked \\
& a &\alpha<1 \checked
\\
\text{mean} \quad  &   a+s \frac{ \alpha}{\gamma-1} \checked
& \gamma >1
\\
\text{variance} \quad  &s^2
\frac{\alpha(\alpha+\gamma-1)}{(\gamma-2)(\gamma-1)^2} \checked
  \hspace{-8em}
&   \gamma>2
\\
\text{skew} \quad  &  \text{not simple}
\\
\text{ex. kurtosis} \quad  &  \text{not simple}
\\
%\text{entropy} \quad  &   \ln \frac{1}{B(\alpha, \gamma)} \Left|\frac{1}{s}\Right|
%  +(1-\alpha) \big[ \psi(\alpha) - \psi(\gamma)\big]
%  \hspace{-8em}
%\\
%& \quad +(\alpha+\gamma) \big[ \psi(\alpha+\gamma) - \psi(\gamma)\big] \checked &  \text{\cite[Eq.~(15)]{Tahmasebi2010}} % Not correct?
%\\
\text{MGF} \quad  &  \text{none}
%\\
%\text{CF} \quad  &  \cdots 
% Possibly expression in terms of second confluence hypergeometric function?
% See wikipedia under F distribution
%\\
%\text{moments} \quad & \frac{B(\alpha+k,\gamma-k)}B(\alpha,\gamma)}
% Only if a =0, s=1. How do moments scale with scale?
\end{align*}
\end{table*}