PropertiesTableBetaLogistic.tex

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\begin{table*}[p]
\caption[Beta-logistic distribution -- Properties]{Properties of the beta-logistic distribution}
\begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad &  \op{BetaLogistic}(x\given \pLoc,\pScale,\alpha,\gamma)  
\checked
\\
\text{PDF}\quad &  
\frac{1}{B(\alpha, \gamma) \Left| \pScale\Right|}
 \frac{e^{-\alpha \frac{x-\pLoc}{\pScale} }} { \Left(1 + e^{-\frac{x-\pLoc}{\pScale}  }\Right)^{\alpha+\gamma} }
\checked 
\\
\text{CDF / CCDF} \quad  &  
 \frac{B\big(\gamma, \alpha ;  ( 1+e^{-\frac{x-\pLoc}{\pScale}} )^{-1}  \big) }{B(\alpha,\gamma)}
& \pScale>0 \, \big/ \, \pScale<0 \checked
 \text{ \cite{\self}} \\
& \qquad  = I\big(  \gamma, \alpha;  ( 1+e^{-\frac{x-\pLoc}{\pScale}} )^{-1}  \big) \checked
\\
\text{parameters}\quad &   \pLoc,\ \pScale,\ \alpha, \gamma  \text{ in } \mathbb{R} \checked \\
& \alpha,\ \gamma >0 \checked
\\
\text{support} \quad &   x \in [-\infty,+\infty] \checked
\\
%\text{median} \quad  &  \cdots
%\\
%\text{mode} \quad  & \cdots
%\\
\text{mean} \quad  &  \pLoc +\pScale [ \psi(\gamma)-\psi(\alpha) ]  \checked %\cite{McDonnald1991} Has difference in scale sign
\\
\text{variance} \quad  & \pScale^2 [\psi_1(\alpha) + \psi_1(\gamma) ] \checked %\cite{McDonnald1991}
\\
\text{skew} \quad  &  \op{sgn}(\lambda)\ \frac{ \psi_2(\gamma) - \psi_2(\alpha)}{[\psi_1(\alpha) + \psi_1(\gamma)]^{3/2}} \checked
\\
\text{ex. kurtosis} \quad  &   \frac{\psi_3(\alpha) + \psi_3(\gamma)}{[\psi_1(\alpha) + \psi_1(\gamma)]^{2}} \checked
\\
%\text{entropy} \quad  & \cdots
%\\
\text{MGF} \quad  &  e^{ \pLoc t}\frac{\Gamma(\alpha-\pScale  t) \Gamma(\gamma+\pScale t)}{\Gamma(\alpha)\Gamma(\gamma)}
& \text{\cite{Johnson1995}} \checked
\\
\text{CF} \quad  &    e^{i \pLoc t} \frac{\Gamma(\alpha+ i \pScale  t) \Gamma(\gamma-i \pScale t)}{\Gamma(\alpha)\Gamma(\gamma)} \checked
\end{align*}
\end{table*}