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Hyperbolic Secant Distribution

docs/tutorial:stats:continuous_hypsecant

Related to the logistic distribution and used in lifetime analysis. Standard form is (defined over all $x$ )

M (t) = sec (\frac{π}{2} t)

\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

where $B_{2 m + 1} (\frac{1}{4})$ is the Bernoulli polynomial of order $2 m + 1$ evaluated at $1/4.$ Thus

μ_{n}^{'} = {0 4 (- 1)^{n /2 - 1} \frac{( 2 π ) ^{n}}{n + 1} B_{n + 1} (\frac{1}{4}) n odd n even

h [X] = lo g (2 π) .

Implementation: scipy.stats.hypsecant