PropertiesTablePearsonIV.tex

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\begin{table*}[t!]
%\addcontentsline{toc}{subsection}{Pearson  IV} 
 \caption[Pearson  IV distribution -- Properties]{Properties of the Pearson  IV distribution}

\begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad & \text{PearsonIV}(x \given a,s, m,v)  
\\
\text{PDF}\quad &    \frac{{}_{2}F_1(-i v, i v ; m;1)  }{|s| B(m-\frac{1}{2}, \frac{1}{2} )} \Left( 1 +\Left( \frac{x-a}{s}\Right)^2 \Right)^{-m}
\\ \notag & \qquad \qquad \qquad \qquad \times \exp \Left\{ - 2 v \arctan \Left( \frac{x-a}{s}\Right) \Right\}
\checked
\\
\text{CDF} \quad  &    \text{PearsonIV}(x\given a,s, m,v)  \\ & \quad  \times \frac{|s|}{2m-1}\Left( i - \frac{x-a}{s}\Right) {}_2F_1\Left(1,m+iv; 2m; \tfrac{2}{i-i\tfrac{x-a}{s}} \Right) \checked
% What if s is negative? Must flip to ccdf as usual!?
% Checked against source. Good to actual check locally.
\\
\text{parameters}\quad &   a,\  s,\  m,\ v \text{ in } \Real
\\ & m>\tfrac{1}{2}
\\
\text{support} \quad &   x \in [-\infty, + \infty]
\\
%\text{median} \quad  &  \cdots
%\\
\text{mode} \quad  & a - \frac{sv}{m}  \checked
\\
\text{mean} \quad  &  a - \frac{sv}{(m-1)} \qquad (m>1) \checked
\\
\text{variance} \quad  & \frac{s^2}{2m-3} (1 + \frac{v^2}{(m-1)^2})  \qquad (m>\frac{3}{2}) \checked
\\
\text{skew} \quad  &  \text{not simple}
\\
\text{ex. kurtosis} \quad  &  \text{not simple}
%\\
%\text{entropy} \quad  & \text{unknown}
%\\
%\text{MGF} \quad  &  \text{unknown}
%\\
%\text{CF} \quad  &  \text{unknown}
\end{align*}
\end{table*}