Hyperbolic Secant Distribution¶
Related to the logistic distribution and used in lifetime analysis. Standard form is (defined over all x )
f(x)=1πsech(x)F(x)=2πarctan(ex)G(q)=log(tan(π2q))
M(t)=sec(π2t)
μ′n=1+(−1)n2π22nn![ζ(n+1,14)−ζ(n+1,34)]={0noddCn/2πn2nneven
where Cm is an integer given by
Cm=(2m)![ζ(2m+1,14)−ζ(2m+1,34)]π2m+122m=4(−1)m−116m2m+1B2m+1(14)
where B2m+1(14) is the Bernoulli polynomial of order 2m+1 evaluated at 1/4. Thus
μ′n={0nodd4(−1)n/2−1(2π)nn+1Bn+1(14)neven
md=mn=μ=0μ2=π24γ1=0γ2=2
h[X]=log(2π).
Implementation: scipy.stats.hypsecant