Hyperbolic secant distribution

Hyperbolic secant distribution

Probability distribution
name =hyperbolic secant
type =density
pdf_

cdf_

parameters ="none"
support =x in (-infty; +infty)!
pdf =frac12 ; operatorname{sech}!left(frac{pi}{2},x ight)!
cdf =frac{2}{pi} arctan!left [exp!left(frac{pi}{2},x ight) ight] !
mean =0
median =0
mode =0
variance =1
skewness =0
kurtosis =2
entropy =4/"π" "K" ;approx 1.16624
mgf =sec(t)! for |t|
char =operatorname{sech}(t)! for |t|
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) is

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

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

:F(x) = frac12 + frac{1}{pi} arctan!left [operatorname{sech}!left(frac{pi}{2},x ight) ight] !:= frac{2}{pi} arctan!left [expleft(frac{pi}{2},x ight) ight] !

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

:F^{-1}(p) = -frac{2}{pi}, operatorname{arcsinh}!left [cot(pi,p) ight] !

where "arcsinh" 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, compared with the standard normal distribution.

References

* W. D. Baten, 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: 284–290.
* 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, [http://cgm.cs.mcgill.ca/~luc/rnbookindex.html "Non-Uniform Random Variate Generation"] , Springer-Verlag, New York. Section IX.7.2.
* Cite journal
author = G.K. Smyth
title = A note on modelling cross correlations: Hyperbolic secant regression
journal = Biometrika
volume = 81
pages = 396–402
year = 1994
url = http://www.statsci.org/smyth/pubs/sech.pdf
doi = 10.1093/biomet/81.2.396

* Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, "Continuous Univariate Distributions", volume 2, ISBN 0-471-58494-0.


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