hyperbolic secant
Probability density function
Cumulative distribution function
Parameters	none
Support	${\displaystyle x\in (-\mathrm{\infty };+\mathrm{\infty })}$
PDF	${\displaystyle \frac{1}{2}\mathrm{sech}\left(\frac{\pi }{2}x\right)}$
CDF	${\displaystyle \frac{2}{\pi }\arctan \left[\exp \left(\frac{\pi }{2}x\right)\right]}$
Quantile	${\displaystyle \frac{2}{\pi }\ln \left[\tan \left(\frac{\pi }{2}p\right)\right]}$
Mean	${\displaystyle 0}$
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle 1}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle 2}$
Entropy	${\displaystyle \ln 4}$
MGF	${\displaystyle \sec (t)}$ for
CF	${\displaystyle \mathrm{sech}(t)}$ for

Inprobability theory andstatistics, thehyperbolic secant distribution is a continuousprobability distribution whoseprobability density function andcharacteristic function are proportional to thehyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocalhyperbolic cosine, and thus this distribution is also called theinverse-cosh distribution.

Generalisation of the distribution gives rise to theMeixner distribution, also known as theNatural Exponential Family - Generalised Hyperbolic Secant orNEF-GHS distribution.

Arandom 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:

${\displaystyle f(x)=\frac{1}{2}\mathrm{sech}\frac{\pi x}{2},}$

where "sech" denotes the hyperbolic secant function.

Thecumulative distribution function (cdf) of the standard distribution is a scaled and shifted version of theGudermannian function,

where "arctan" is theinverse (circular) tangent function.

Johnson et al. (1995)^[1]: 147 places this distribution in the context of a class of generalized forms of thelogistic distribution, but use a different parameterisation of the standard distribution compared to that here. Ding (2014)^[2] shows three occurrences of the Hyperbolic secant distribution in statistical modeling and inference.

The hyperbolic secant distribution shares many properties with the standardnormal distribution: it is symmetric with unitvariance and zeromean,median andmode, and its probability density function is proportional to its characteristic function. However, the hyperbolic secant distribution isleptokurtic; 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 thelogistic distribution are special cases of theChampernowne distribution, which has exponential tails.

The inverse cdf (or quantile function) for a uniform variate 0 ≤ p < 1 is

${\displaystyle F^{-1}(p)=-\frac{2}{\pi }\mathrm{arsinh}\left[\cot (\pi p)\right],}$

${\displaystyle =\frac{2}{\pi }\ln \left[\tan \left(\frac{\pi }{2}p\right)\right].}$

where "arsinh" is theinverse hyperbolic sine function and "cot" is the(circular) cotangent function.

Considering the (scaled) sum of${\displaystyle r}$independent and identically distributed hyperbolic secant random variables:

${\displaystyle X=\frac{1}{\sqrt{r}}(X_1+X_2+...+X_r)}$

then in the limit${\displaystyle r\to \mathrm{\infty }}$ the distribution of${\displaystyle X}$ will tend to the normal distribution${\displaystyle N(0,1)}$, in accordance with thecentral limit theorem.

This allows a convenient family of distributions to be defined with properties intermediate between the hyperbolic secant and the normal distribution, controlled by the shape parameter${\displaystyle r}$, which can be extended to non-integer values via thecharacteristic function

${\displaystyle \phi (t)=(\mathrm{sech}(t/\sqrt{r}){)}^r}$

Moments can be readily calculated from the characteristic function. The excesskurtosis is found to be${\displaystyle 2/r}$.

The distribution (and its generalisations) can also trivially be shifted and scaled in the usual way to give a correspondinglocation-scale family:

${\displaystyle f(x)=\frac{1}{2\sigma }\mathrm{sech}\left(\frac{\pi }{2}\frac{x-\mu }{\sigma }\right)}$

Askewed form of the distribution can be obtained by multiplying by the exponential and normalising, to give the distribution

${\displaystyle f(x)=\cos \theta \frac{e^{\theta x}}{2\cosh (\frac{\pi x}{2})}}$

where the parameter value${\displaystyle \theta =0}$ corresponds to the original distribution.

TheChampernowne distribution has an additional parameter to shape the core or wings.

Allowing all four of the adjustments above gives distribution with four parameters, controlling shape, skew, location, and scale respectively, called either theMeixner distribution^[3] afterJosef Meixner who first investigated the family, or theNEF-GHS distribution (Natural exponential family - Generalised Hyperbolic Secant distribution).

Infinancial mathematics the Meixner distribution has been used to model non-Gaussian movement of stock-prices, with applications including the pricing ofoptions.

Losev (1989) has studied independently the asymmetric (skewed) curve${\displaystyle h(x)=\frac{1}{\exp (-ax)+\exp (bx)}}$, which uses just two parameters${\displaystyle a,b}$. In it,${\displaystyle a}$ is a measure of left skew and${\displaystyle b}$ a measure of right skew, in case the parameters are both positive. They have to be both positive or negative, with${\displaystyle a=b}$ being the hyperbolic secant - and therefore symmetric - and${\displaystyle h(x{)}^r}$ being its further reshaped form.^[4]

The normalising constant is as follows:

${\displaystyle \frac{a+b}{\pi }\sin \left(\frac{b\pi }{a+b}\right)}$

which reduces to${\displaystyle \frac{2a}{\pi }}$ for the symmetric version.

Furthermore, for the symmetric version,${\displaystyle a}$ can be estimated as${\displaystyle a=\frac{\pi }{2\sigma }}$.

• Baten, W. D. (1934)."The probability law for the sum ofn independent variables, each subject to the law${\displaystyle (2h{)}^{-1}\mathrm{sech}(\pi x/2h)}$".Bulletin of the American Mathematical Society.40 (4):284–290.doi:10.1090/S0002-9904-1934-05852-X.
• Talacko, J. (1956). "Perks' distributions and their role in the theory of Wiener's stochastic variables".Trabajos de Estadistica.7 (2):159–174.doi:10.1007/BF03003994.S2CID 120569210.
• Devroye, Luc (1986).Non-uniform random variate generation. New York: Springer-Verlag. Section IX.7.2.
• Smyth, G.K. (1994)."A note on modelling cross correlations: Hyperbolic secant regression"(PDF).Biometrika.81 (2):396–402.doi:10.1093/biomet/81.2.396.
• Matthias J. Fischer (2013),Generalized Hyperbolic Secant Distributions: With Applications to Finance, Springer.ISBN 3642451381.Google Books

• Continuous distributions

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