Hyperbolic secant distribution

hyperbolic secant
Probability density function

Cumulative distribution function

Parameters none
Support x \in (-\infty; +\infty)\!
PDF \frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!
CDF \frac{2}{\pi} \arctan\!\left[\exp\!\left(\frac{\pi}{2}\,x\right)\right]\!
Mean 0
Median 0
Mode 0
Variance 1
Skewness 0
Ex. kurtosis 2
Entropy 4/π K \;\approx 1.16624
MGF \sec(t)\! for |t|<\frac{\pi}2\!
CF \operatorname{sech}(t)\! for |t|<\frac{\pi}2\!

In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the inverse hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution.

Explanation

A random variable follows a hyperbolic secant distribution if its probability density function (pdf) can be related to the following standard form of density function by a location and shift transformation:

f(x) = \frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\! ,

where "sech" denotes the hyperbolic secant function. The cumulative distribution function (cdf) of the standard distribution is

F(x) = \frac12 + \frac{1}{\pi} \arctan\!\left[\operatorname{sinh}\!\left(\frac{\pi}{2}\,x\right)\right] \! ,
= \frac{2}{\pi} \arctan\!\left[\exp\left(\frac{\pi}{2}\,x\right)\right] \! .

where "arctan" is the inverse (circular) tangent function. The inverse cdf (or quantile function) is

F^{-1}(p) = -\frac{2}{\pi}\, \operatorname{arsinh}\!\left[\cot(\pi\,p)\right] \! ,
= \frac{2}{\pi}\, \ln\!\left[\tan\left(\frac{\pi}{2}\,p\right)\right] \! .

where "arsinh" is the inverse hyperbolic sine function and "cot" is the (circular) cotangent function.

The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its pdf is proportional to its characteristic function. However, the hyperbolic secant distribution is leptokurtic; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution.

Johnson et al. (1995, p147) place this distribution in the context of a class of generalised forms of the logistic distribution, but use a different parameterisation of the standard distribution compared to that here. Ding (2014) shows three occurrences of the Hyperbolic secant distribution in statistical modeling and inference.

References