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The stretched exponential distribution may be useful for fitting
fat-tailed empirical distributions.
The stretched exponential has a similar functional form as the Weibull
distribution, and the Weibull is confusingly sometimes referred to as
a ``stretched exponential distribution'' in the literature, but they
are not the same. (See the \eslmod{weibull} module.)
\begin{tabular}{lcll}\hline
Variate & $x$ & \ccode{double} & $\mu \leq x < \infty$ \\
Location & $\mu$ & \ccode{double} & $-\infty < \mu < \infty$\\
Scale & $\lambda$ & \ccode{double} & $\lambda > 0$ \\
Shape & $\tau$ & \ccode{double} & $\tau > 0$ \\ \hline
\end{tabular}
The probability density function (PDF) is:
\begin{equation}
P(X=x) = \frac{\lambda \tau}{\Gamma(\frac{1}{\tau})} e^{- [\lambda(x-\mu)]^{\tau}}
\label{eqn:stretchexp_pdf}
\end{equation}
The cumulative distribution function (CDF) does not have an analytical
expression. It is obtained from the integral:
\begin{eqnarray*}
P(X \leq x) & = & \int_{\mu}^{x} P(X=z) dz\\
& = & \frac{\lambda \tau}{\Gamma(\frac{1}{\tau})} \int_\mu^{x} e^{- [\lambda(z-\mu)]^{\tau}} dz\\
\label{eqn:stretchexp_cdf1}
\end{eqnarray*}
By change-of-variables $t = [\lambda(z-\mu)]^{\tau}$,
$t^{\frac{1}{\tau}} = \lambda(z-\mu)$, $dz = \frac{1}{\lambda \tau}
t^{\frac{1}{\tau}-1} dt$, this can be rewritten as:
\[
P(X \leq x) = \frac{1}{\Gamma(\frac{1}{\tau})}
\int_0^{[\lambda(x-\mu)]^{\tau}} e^{-t} t^{\frac{1}{\tau}-1} dt
\]
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