PropertiesTableGenBeta.tex

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\begin{table*}[tp]
\caption[Generalized beta distribution -- Properties]{Properties of the generalized beta distribution}
 \begin{align*}
 \text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{name} \quad & \text{GenBeta}(x\given a,s,\alpha, \gamma,\beta) 
\\
\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)^{\gamma -1}
\checked
\hspace{-8em}
\\
\text{CDF / CCDF} \quad  &   \frac{B\Left( \alpha,\gamma; (\tfrac{x-a}{s})^\beta  \Right)}{B(\alpha,\gamma)}
\checked
% McDonald1984 table 3.1 GB1
& \tfrac{\beta}{s} >0 \,\big/ \, \tfrac{\beta}{s} <0
\\ & \quad = I\Left(  \alpha,\gamma; (\tfrac{x-a}{s})^\beta \Right) \checked
\\ 
\text{parameters}\quad &   a,\ s,\ \alpha,\ \gamma,\ \beta, \text{ in } \Real, \\ &  \alpha,\gamma\geq0
\\
\text{support} \quad 
&   \ x \in [a,a+s],  \quad & 0<s,\ 0<\beta  \checked
 \\ 	 		 & \ x\in[a+s,a],  \quad & s<0,\ 0<\beta   \checked
 \\  			 & \ x\in[a+s,+\infty], \quad  & 0<s,\ \beta<0  \checked
 \\  			& \ x\in[-\infty,a+s], \quad & s<0,\ \beta<0 \checked
\\
%\text{median} \quad  &  \cdots
%\\
%\text{mode} \quad  & \cdots
%\\
\text{mean} \quad  &   a+ \frac{s B(\alpha+\tfrac{1}{\beta},\gamma) }{B(\alpha,\gamma)}  \quad & \alpha+\tfrac{1}{\beta} >0
\checked
\\
\text{variance} \quad  & \frac{s^2 B(\alpha+\tfrac{2}{\beta},\gamma) }{B(\alpha,\gamma)} -\frac{s^2 B(\alpha+\tfrac{1}{\beta},\gamma)^2 }{B(\alpha,\gamma)^2} \checked
\\
\text{skew} \quad  &   \text{not simple}
\\
\text{ex. kurtosis} \quad  &   \text{not simple}
\\
% \text{entropy} \quad  & \cdots
% \\
\text{MGF} \quad  &  \text{none} % Can't exist because beta mgf doesn't exist
\\
% \text{CF} \quad  &  \cdots
% \\
E(X^h) \quad & \frac{s^h B(\alpha+\tfrac{h}{\beta},\gamma) }{B(\alpha,\gamma)}  \qquad  & \!\!a=0,\ \alpha+\tfrac{h}{\beta} >0
\ \text{\cite{McDonald1984}} \checked
%{Normalize style, use similar style in beta properties table.}
\end{align*}
\end{table*}