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: $P(X \leq x) = \frac{1}{\Gamma(\frac{1}{\tau})} \int_0^{[\lambda(x-\mu)]^{\tau}} e^{-t} t^{\frac{1}{\tau}-1} dt$