1 2\begin{tabular}{lcll}\hline 3Variate & $x$ & \ccode{double} & $\mu \leq x < \infty$ \\ 4Location & $\mu$ & \ccode{double} & $-\infty < \mu < \infty$\\ 5Scale & $\lambda$ & \ccode{double} & $\lambda > 0$ \\ 6Shape & $\tau$ & \ccode{double} & $\tau > 0$ \\ \hline 7\end{tabular} 8 9The probability density function (PDF) is: 10 11\begin{equation} 12P(X=x) = \frac{\lambda^{\tau}}{\Gamma(\tau)} (x-\mu)^{\tau-1} e^{-\lambda (x - \mu)} 13\label{eqn:gamma_pdf} 14\end{equation} 15 16The cumulative distribution function (CDF) does not have an analytical 17expression. It is calculated numerically, using the incomplete Gamma 18function (\ccode{esl\_stats\_IncompleteGamma()}). 19 20The ``standard Gamma distribution'' has $\mu = 0$, $\lambda = 1$. 21 22\subsection{Sampling} 23 24 25 26\subsection{Parameter estimation} 27 28\subsubsection{Complete data; known location} 29 30We usually know the location $\mu$. It is often 0, or in the case of 31fitting a gamma density to a right tail, we know the threshold $\mu$ 32at which we truncated the tail. 33 34Given a complete dataset of $N$ observed samples $x_i$ ($i=1..N$) and 35a \emph{known} location parameter $\mu$, maximum likelihood estimation 36of $\lambda$ and $\tau$ is performed by first solving this rootfinding 37equation for $\hat{\tau}$ by binary search: 38 39\begin{equation} 40 \log \hat{\tau} 41 - \Psi(\hat{\tau}) 42 - \log \left[ \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu) \right] 43 + \frac{1}{N} \sum_{i=1}^N \log (x_i - \mu) 44\label{eqn:gamma_tau_root} 45\end{equation} 46 47then using that to obtain $\hat{\lambda}$: 48 49\begin{equation} 50\hat{\lambda} = \frac{N \hat{\tau}} {\sum_{i=1}^{N} (x_i - \mu)} 51\end{equation} 52 53Equation~\ref{eqn:gamma_tau_root} decreases as $\tau$ increases. 54