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.

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.

References

• Baten, W. D. (1934). "The probability law for the sum of n independent variables, each subject to the law $(2h)^{-1} \operatorname{sech}(\pi x/2h)$". Bulletin of the American Mathematical Society 40 (4): 284–290. doi:10.1090/S0002-9904-1934-05852-X. 
• J. Talacko, 1956, "Perks' distributions and their role in the theory of Wiener's stochastic variables", Trabajos de Estadistica 7:159–174.
• Luc Devroye, 1986, Non-Uniform Random Variate Generation, Springer-Verlag, New York. Section IX.7.2.
• G.K. Smyth (1994). "A note on modelling cross correlations: Hyperbolic secant regression". Biometrika 81 (2): 396–402. doi:10.1093/biomet/81.2.396. http://www.statsci.org/smyth/pubs/sech.pdf. 
• Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, Continuous Univariate Distributions, volume 2, ISBN 0-471-58494-0.