Probability density function

A random 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:

${\displaystyle f(x)={\frac {1}{2}}\;\operatorname {sech} \!\left({\frac {\pi }{2}}\,x\right)\!,}$

where "sech" denotes the hyperbolic secant function.

Cumulative distribution function

The cumulative distribution function (cdf) of the standard distribution is a scaled and shifted version of the Gudermannian function,

${\displaystyle F(x)={\frac {1}{2}}+{\frac {1}{\pi }}\arctan \!\left[\operatorname {sinh} \!\left({\frac {\pi }{2}}\,x\right)\right]\!,}$

${\displaystyle ={\frac {2}{\pi }}\arctan \!\left[\exp \left({\frac {\pi }{2}}\,x\right)\right]\!.}$

where "arctan" is the inverse (circular) tangent function.

Johnson et al. (1995)^[1]: 147 places this distribution in the context of a class of generalized forms of the logistic 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.

Convolution

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 ${\displaystyle r\to \infty }$ the distribution of ${\displaystyle X}$ will tend to the normal distribution ${\displaystyle N(0,1)}$, in accordance with the central 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 the characteristic function

${\displaystyle \varphi (t)={\big (}\operatorname {sech} (t/{\sqrt {r}}){\big )}^{r}}$

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

Skew

A skewed form of the distribution can be obtained by multiplying by the exponential ${\displaystyle e^{\theta x},|{\theta }|<{\pi }/2}$ and normalising, to give the distribution

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

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

Location and scale

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

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

Meixner distribution

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

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

Related distribution

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 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 ${\displaystyle a={\frac {\pi }{2\sigma }}}$.