Half-Logistic DistributionΒΆ
In the limit as \(c\rightarrow\infty\) for the generalized half-logistic we have the half-logistic defined over \(x\geq0.\) Also, the distribution of \(\left|X\right|\) where \(X\) has logistic distribution.
\begin{eqnarray*} f\left(x\right) & = & \frac{2e^{-x}}{\left(1+e^{-x}\right)^{2}}=\frac{1}{2}\mathrm{sech}^{2}\left(\frac{x}{2}\right)\\
F\left(x\right) & = & \frac{1-e^{-x}}{1+e^{-x}}=\tanh\left(\frac{x}{2}\right)\\
G\left(q\right) & = & \log\left(\frac{1+q}{1-q}\right)=2\mathrm{arctanh}\left(q\right)\end{eqnarray*}
\[M\left(t\right)=1-t\psi_{0}\left(\frac{1}{2}-\frac{t}{2}\right)+t\psi_{0}\left(1-\frac{t}{2}\right)\]
where \(\psi_m\) is the polygamma function \(\psi_m(z) = \frac{d^{m+1}}{dz^{m+1}} \log(\Gamma(z))\).
\[\mu_{n}^{\prime}=2\left(1-2^{1-n}\right)n!\zeta\left(n\right)\quad n\neq1\]
\begin{eqnarray*} \mu_{1}^{\prime} & = & 2\log\left(2\right)\\
\mu_{2}^{\prime} & = & 2\zeta\left(2\right)=\frac{\pi^{2}}{3}\\
\mu_{3}^{\prime} & = & 9\zeta\left(3\right)\\
\mu_{4}^{\prime} & = & 42\zeta\left(4\right)=\frac{7\pi^{4}}{15}\end{eqnarray*}
\begin{eqnarray*} h\left[X\right] & = & 2-\log\left(2\right)\\ & \approx & 1.3068528194400546906.\end{eqnarray*}
Implementation: scipy.stats.halflogistic