PropertiesTableUnitGamma.tex

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\begin{table*}[t!]
\caption[Unit gamma distribution -- Properties]{Properties of the unit gamma distribution}
\begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad & \text{UnitGamma}(x\given a,s,\alpha,\beta) 
\\
\text{PDF}\quad &  \frac{1}{\Gamma(\alpha)} \Left|\frac{\beta}{ s}\Right|
\Left(\frac{x-a}{s}\Right)^{\beta-1} \Left(-\beta \ln   \frac{x-a}{s} \Right)^{\alpha-1} \checked
\\
\text{CDF} \big/ \text{CCDF}  \quad  & 1-Q\Left(\alpha, -\beta \ln \tfrac{x-a}{s} \Right)
& \tfrac{\beta}{s}  >0 \ \big{/} \ \tfrac{\beta}{s}<0   \checked
\\
\text{parameters}\quad &   a, s,\alpha,\beta \text{ in } \Real  ,\  \alpha, \beta>0
\\
\text{support}\checked \quad 
	&   [a,a+s] ,\ s>0,\ \beta>0 \\
	&  [a+s,a], \ s<0,\ \beta>0 \\
	& [a+s,+\infty]  s>0,\ \beta<0  \\
 	& [-\infty,a+s], s<0,\ \beta<0 \\
%\text{median} \quad  &  \cdots
%\\
%\text{mode} \quad  & a+s \exp \Left(\tfrac{1-\alpha}{1-\beta} \Right)  \text{ if } \alpha,\beta>0 \text{ or } \alpha,\beta<0 \text{ (antimode)} \hspace{-4em}
%  ????
%\\
\text{mean} \quad  &  a + s \Left(\tfrac{\beta}{\beta+1} \Right)^{\alpha} \checked
\\
\text{variance} \quad  & s^2 \Left(\tfrac{\beta}{\beta+2} \Right)^{\alpha} - s^2 \Left(\tfrac{\beta}{\beta+1} \Right)^{2\alpha} \checked
\\
\text{skew} \quad  &   \text{not simple}
\\
\text{ex. kurtosis} \quad  &   \text{not simple}
\\
%\text{entropy} \quad  & \cdots
%\\
%\text{MGF} \quad  &  \cdots
%\\
%\text{CF} \quad  &  \cdots
%\\
E(X^h) \quad & \Left(\tfrac{\beta}{\beta+h} \Right)^{\alpha} \qquad  & a=0 \
\text{\cite{Grassia1977}} \checked
\end{align*}
\end{table*}