PropertiesTableAmoroso.tex

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\begin{table*}[pt!]

\caption[Amoroso distribution -- Properties]{Properties of the Amoroso distribution}
%\addcontentsline{toc}{subsection}{Amoroso} 

\begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad &  \op{Amoroso}(x\given a, \theta, \alpha, \beta)  \checked
\\
\text{PDF} \quad &
\frac{1}{\Gamma(\alpha)} 
\Left|\frac{\beta}{\theta}\Right|
\Left(\frac{x-a}{\theta}\Right)^{\alpha \beta -1}
\exp \Left\{
-  \Left(\frac{x-a}{\theta}\Right)^{\beta}
\Right\}
\checked
\hspace{-8em}
\\ 
\text{CDF / CCDF } \quad  &    1-Q\Left(\alpha, \Left(\tfrac{x - a }{\theta}\Right)^{\beta}\Right) 
\checked & \tfrac{\theta}{\beta}>0 \, \big/ \,  \tfrac{\theta}{\beta}<0
%\\
%& Q(\alpha, \Left(\tfrac{x - a }{\theta}\Right)^{\beta}) 
%& \tfrac{\beta}{\theta}<0
\\
\text{parameters}\quad &   a,\ \theta,\ \alpha,\ \beta\  \text{in } \Real, \ \alpha>0	\checked
\\
\text{support} \quad &     x \geq a &  \theta > 0 \checked
\\
&   x\leq a  &  \theta < 0 	\checked
\\
%\text{median} \quad  &  \cdots
%\\
\text{mode} \quad&   a+ \theta (\alpha-\tfrac{1}{\beta})^{\frac{1}{\beta}}  \checked
& \alpha \beta  \geq 1		\checked
\\ & a & \alpha \beta  \le 1
\\
\text{mean} \quad& a  + \theta \frac{\Gamma(\alpha+\frac{1}{\beta})}{\Gamma(\alpha)}  \checked
& \alpha + \tfrac{1}{\beta} \geq 0
\\
\text{variance}  \quad&   \theta^2 \Left[  \frac{\Gamma(\alpha+\frac{2}{\beta})}{\Gamma(\alpha)}  - 
\frac{\Gamma(\alpha+\frac{1}{\beta})^2}{\Gamma(\alpha)^2}    \Right] \checked
& \alpha + \tfrac{2}{\beta} \geq 0
\\
\text{skew} \quad  &  \op{sgn}(\tfrac{\beta}{\theta})\ \Left[  \tfrac{\Gamma(\alpha+\frac{3}{\beta})}{\Gamma(\alpha)} - 3 \tfrac{\Gamma(\alpha+\frac{2}{\beta})\Gamma(\alpha+\frac{1}{\beta})}{\Gamma(\alpha)^2}    + 2  \tfrac{\Gamma(\alpha+\frac{1}{\beta})^3}{\Gamma(\alpha)^3}   \Right]
 \hspace{-3em}
 \\ & \qquad \qquad \qquad \qquad \Big /
 \Left[  \tfrac{\Gamma(\alpha+\frac{2}{\beta})}{\Gamma(\alpha)}  - 
\tfrac{\Gamma(\alpha+\frac{1}{\beta})^2}{\Gamma(\alpha)^2}    \Right]^{3/2}
\hspace{-3em}
\checked
\\
\text{ex. kurtosis} \quad  &  
 \bigg[  \tfrac{\Gamma(\alpha+\frac{4}{\beta})}{\Gamma(\alpha)} 
 - 4 \tfrac{\Gamma(\alpha+\frac{3}{\beta})\Gamma(\alpha+\frac{1}{\beta})}{\Gamma(\alpha)^2}    
 + 6 \tfrac{\Gamma(\alpha+\frac{2}{\beta})\Gamma(\alpha+\frac{1}{\beta})^2}{\Gamma(\alpha)^3}    
\hspace{-3em}
\checked
 \\ & \qquad 
 -3  \tfrac{\Gamma(\alpha+\frac{1}{\beta})^4}{\Gamma(\alpha)^4}   \bigg]
 \Big /
 \Left[  \tfrac{\Gamma(\alpha+\frac{2}{\beta})}{\Gamma(\alpha)}  - 
\tfrac{\Gamma(\alpha+\frac{1}{\beta})^2}{\Gamma(\alpha)^2}    \Right]^{2}
-3 
\hspace{-3em}
\\
\text{entropy} \quad& 
\ln \frac{|\theta| \Gamma(\alpha)}{|\beta|} +\alpha + \Left( \sfrac{1}{\beta} - \alpha\Right) \psi(\alpha) \checked &
\text{\cite{Dadpay2007}}
%\\
%\text{MGF} \quad  &  \cdots
%\\
%\text{CF} \quad  &  \cdots
\end{align*}
\end{table*}