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 reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution. Generalisation of the distribution gives rise to the Meixner distribution, also known as the Natural Exponential Family - Generalised Hyperbolic Secant or NEF-GHS distribution.
Definitions
Probability density function A random variable follows a hyperbolic secant distribution if its probability density function can be related to the following standard form of density function by a location and shift transformation:
f ( x ) = 1 2 sech π x 2 , {\displaystyle f(x)={\frac {1}{2}}\operatorname {sech} {\frac {\pi x}{2}},}
where "sech" denotes the hyperbolic secant function.
Cumulative distribution function The cumulative distribution function (cdf) of the standard distribution is a scaled and shifted version of the Gudermannian function,
F ( x ) = 1 2 + 1 π arctan ( sinh π x 2 ) = 2 π arctan ( exp π x 2 ) . {\displaystyle {\begin{aligned}F(x)&={\frac {1}{2}}+{{\frac {1}{\pi }}\arctan }{\Bigl (}{\operatorname {sinh} {\frac {\pi x}{2}}}{\Bigr )}\\[8mu]&={{\frac {2}{\pi }}\arctan }{\Bigl (}{\exp {\frac {\pi x}{2}}}{\Bigr )}.\end{aligned}}}
where "arctan" is the inverse (circular) tangent function. Johnson et al. (1995) places this distribution in the context of a class of generalized 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.
Properties 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 probability density function 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. Both the hyperbolic secant distribution and the logistic distribution are special cases of the Champernowne distribution, which has exponential tails. The inverse cdf (or quantile function) for a uniform variate 0 ≤ p < 1 is
F − 1 ( p ) = − 2 π arsinh [ cot ( π p ) ] , {\displaystyle F^{-1}(p)=-{\frac {2}{\pi }}\,\operatorname {arsinh} \!\left[\cot(\pi \,p)\right]\!,}
= 2 π ln [ tan ( π 2 p ) ] . {\displaystyle ={\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.
Generalisations
Convolution Considering the (scaled) sum of r {\displaystyle r} independent and identically distributed hyperbolic secant random variables:
X = 1 r ( X 1 + X 2 + . . . + X r ) {\displaystyle X={\frac {1}{\sqrt {r}}}\;(X_{1}+X_{2}+\;...\;+X_{r})}
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