Hyperbolic secant distribution

hyperbolic secant
Probability density function
Cumulative distribution function
Parameters	none
Support	$x\in (-\infty ;+\infty )\!$
PDF	${\frac 12}\;\operatorname {sech}\!\left({\frac {\pi }{2}}\,x\right)\!$
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)={\frac 12}\;\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)={\frac 12}+{\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

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" (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.
Ding, P. 2014, "Three Occurrences of the Hyperbolic-Secant Distribution", The American Statistician, 68, 32-35.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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