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Hyperbolic Secant Distribution¶
Related to the logistic distribution and used in lifetime analysis. Standard form is (defined over all \(x\) )
\begin{eqnarray*} f\left(x\right) & = & \frac{1}{\pi}\mathrm{sech}\left(x\right)\\
F\left(x\right) & = & \frac{2}{\pi}\arctan\left(e^{x}\right)\\
G\left(q\right) & = & \log\left(\tan\left(\frac{\pi}{2}q\right)\right)\end{eqnarray*}
\[M\left(t\right)=\sec\left(\frac{\pi}{2}t\right)\]
\begin{eqnarray*} \mu_{n}^{\prime} & = & \frac{1+\left(-1\right)^{n}}{2\pi2^{2n}}n!\left[\zeta\left(n+1,\frac{1}{4}\right)-\zeta\left(n+1,\frac{3}{4}\right)\right]\\
& = & \left\{
\begin{array}{cc}
0 & n \text{ odd}\\
C_{n/2}\frac{\pi^{n}}{2^{n}} & n \text{ even}
\end{array}
\right.\end{eqnarray*}
where \(C_{m}\) is an integer given by
\begin{eqnarray*} C_{m} & = & \frac{\left(2m\right)!\left[\zeta\left(2m+1,\frac{1}{4}\right)-\zeta\left(2m+1,\frac{3}{4}\right)\right]}{\pi^{2m+1}2^{2m}}\\
& = & 4\left(-1\right)^{m-1}\frac{16^{m}}{2m+1}B_{2m+1}\left(\frac{1}{4}\right)\end{eqnarray*}
where \(B_{2m+1}\left(\frac{1}{4}\right)\) is the Bernoulli polynomial of order \(2m+1\) evaluated at \(1/4.\) Thus
\[\begin{split}\mu_{n}^{\prime}=\left\{
\begin{array}{cc}
0 & n \text{ odd}\\
4\left(-1\right)^{n/2-1}\frac{\left(2\pi\right)^{n}}{n+1}B_{n+1}\left(\frac{1}{4}\right) & n \text{ even}
\end{array}
\right.\end{split}\]
\begin{eqnarray*} m_{d}=m_{n}=\mu & = & 0\\
\mu_{2} & = & \frac{\pi^{2}}{4}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & 2\end{eqnarray*}
\[h\left[X\right]=\log\left(2\pi\right).\]
Implementation: scipy.stats.hypsecant