Hyperbolic secant distribution
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Probability density function  | |||
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Cumulative distribution function  | |||
| Parameters | none | ||
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| Support | |||
| CDF | |||
| Mean | |||
| Median | |||
| Mode | |||
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| Excess kurtosis | |||
| Entropy | 4/π K | ||
| MGF | for | ||
| CF | for | ||
![{\displaystyle {\frac {2}{\pi }}\arctan \!\left[\exp \!\left({\frac {\pi }{2}}\,x\right)\right]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7464c4d560a91320ee31357daccac2ea545a486f)
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:
where "sech" denotes the hyperbolic secant function. The cumulative distribution function (cdf) of the standard distribution is
![{\displaystyle F(x)={\frac {1}{2}}+{\frac {1}{\pi }}\arctan \!\left[\operatorname {sinh} \!\left({\frac {\pi }{2}}\,x\right)\right]\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e395f5df13f450fafb9616b8da410494091da39f)
![{\displaystyle ={\frac {2}{\pi }}\arctan \!\left[\exp \left({\frac {\pi }{2}}\,x\right)\right]\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88d70856f9e81af923911fdd8f829285225d8ce6)
where "arctan" is the inverse (circular) tangent function. The inverse cdf (or quantile function) is
![{\displaystyle F^{-1}(p)=-{\frac {2}{\pi }}\,\operatorname {arsinh} \!\left[\cot(\pi \,p)\right]\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4e00fec9c7e8180254e0075e00abfe66c92a43)
![{\displaystyle ={\frac {2}{\pi }}\,\ln \!\left[\tan \left({\frac {\pi }{2}}\,p\right)\right]\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a987b33fad985cdae6ac309bfb06f3ca0d7b663)
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 ". 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" (PDF). Biometrika. 81 (2): 396–402. 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.
