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: where "sech" denotes the hyperbolic secant function. The cumulative distribution function of the standard distribution is where "arctan" is the inverse tangent function. The inverse cdf is where "arcsinh" is the inverse hyperbolic sine function and "cot" is the 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. 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 shows three occurrences of the Hyperbolic secant distribution in statistical modeling and inference.
Generalisations
Convolution
Considering the sum of independent and identically distributed hyperbolic secant random variables: then in the limit the distribution of will tend to the normal distribution, 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, which can be extended to non-integer values via the characteristic function Moments can be readily calculated from the characteristic function. The excess kurtosis is found to be.
Skew
A skewed form of the distribution can be obtained by multiplying by the exponential and normalising, to give the distribution where the parameter value corresponds to the original distribution.
Location and scale
The distribution can also trivially be shifted and scaled in the usual way to give a corresponding location-scale family
All of the above
Allowing all four of the adjustments above gives distribution with four parameters, controlling shape, skew, location, and scale respectively, called either the Meixner distribution after Josef Meixner who first investigated the family, or the NEF-GHS distribution. Losev has studied independantly the asymmetric curve which uses just two parameters. They have to be both both positive or negative, with being the secant, and being its further reshaped form.. In financial mathematics the Meixner distribution has been used to model non-Gaussian movement of stock-prices, with applications including the pricing of options.
