The Hyperbolic Secant distribution (also called the inverse-cosh distribution) is a location-scale family with unique properties that make it suitable for some limited applications. In this blog post, we’ll take a closer look at what the Hyperbolic Secant distribution is, its properties, and how it can be used.
What is the hyperbolic secant distribution?
The Hyperbolic Secant distribution, also known as the sech distribution, is derived from the hyperbolic function. It is a member of the exponential family which bears similarities to the normal distribution; both distributions have a density proportional to their characteristic functions, but the hyperbolic secant has heavier tails .
The distribution’s density is proportional to the hyperbolic secant function, the reciprocal of the hyperbolic cosine function defined by :
cosh(x) = 0.5 [ex + e-x]
The probability density function (PDF) of the hyperbolic secant can be expressed as:
Where μ ∈ ℝ and
The Hyperbolic Secant is tractable — calculation of the distribution at any point takes polynomial-time. The sample mean and median are equally efficient estimators for the population distribution .
Generalization of the hyperbolic secant distribution results in the Meixner distribution.
Similarity to the normal distribution
The hyperbolic secant distribution and normal distribution are both symmetric with unit variance and zero mean. In addition, both distributions have density proportional to their characteristic functions. However, unlike the normal distribution, the Hyperbolic Secant distribution has heavier tails (i.e., it is leptokurtic). The distribution has a positive kurtosis of 2, which means it has fewer outliers than the normal distribution.
Use of the hyperbolic secant distribution
One of the advantages of the Hyperbolic Secant distribution is that it is easy to work with mathematically. The density function can be computed rapidly, which is useful for simulations and optimization problems. Furthermore, it is symmetric about its mean, and the median equals the mean, which eases the interpretation of the results. Despite these advantages, this distribution isn’t used a lot for practical purposes. But the likelihood of extreme values makes it useful in modeling phenomena that have a wide range of values, such as stock prices or weather patterns.
The Hyperbolic Secant distribution has been applied in various fields including finance, engineering, and physics. In finance, it has been used to model volatility in financial markets, and its robustness to heavy tails and skewness makes it ideal for capturing extreme market outcomes. In engineering, the distribution has been used to model failure times of mechanical systems. In physics, it has been used in studies of particle physics to describe the distribution of energies or momenta.
The Hyperbolic Secant distribution may not be seen as often as some other distributions due to its isolation from well-known statistical models , but it can be a useful tool for modeling phenomena with wide ranges of values and heavy tails. Its easy computation, symmetric and platykurtic properties, and application in various fields make it worth considering for statistical analysis. As you continue to learn and apply probability theory, it is important to have a diverse set of tools in your statistical toolbox. The Hyperbolic Secant distribution is one such tool.
 IkamusumeFan, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons
 M. J. Fischer, Generalized Hyperbolic Secant Distributions, 1
SpringerBriefs in Statistics, DOI: 10.1007/978-3-642-45138-6_1
 Stat 5601 (Geyer) Hyperbolic Secant Distribution. Retrieved December 18, 2021 from: https://www.stat.umn.edu/geyer/old02/5601/examp/hsec.html