Hyperbolic Secant Distribution

The hyperbolic secant distribution, also known as the inverse hyperbolic cosine distribution, is a continuous probability distribution with lighter tails than the normal distribution. It is used in finance and telecommunications to model certain types of data.

Definition

The hyperbolic secant distribution has a location parameter $a$ and a scale parameter $b$ . Its probability density function (PDF) is given by:

f (x) = \frac{1}{2 b} sech (\frac{π}{2} \frac{x - a}{b})

The cumulative distribution function (CDF) is:

F (x) = \frac{2}{π} \arctan (\exp (\frac{π}{2} \frac{x - a}{b}))

where $sech (x)$ is the hyperbolic secant function. The domain of the distribution is all real numbers, and the parameters must satisfy $b > 0$ .

Applications

The distribution is used in financial modeling to analyze asset returns that exhibit symmetric, heavy-tailed behavior.
In telecommunications, it serves as a model for signal noise characteristics.
Statistical physics applications employ this distribution to model particle behavior and energy distributions.
The distribution provides a framework for analyzing measurement errors in precision systems.

Properties

Statistical Properties

Property	Value
Mean	$a$
Median	$a$
Mode	$a$
Variance	$\frac{π^{2}}{4} b^{2}$
Skewness	0
Excess Kurtosis	2
Entropy	$\ln (4 b)$

Notable properties include the following characteristics:

The distribution is symmetric around the location parameter $a$ . Additionally, it exhibits scale-invariance up to location and scale parameters. The distribution also possesses the property of infinite divisibility.

Relationships to Other Distributions

The hyperbolic secant distribution has several important relationships with other probability distributions:

When two independent hyperbolic secant random variables are summed, the result follows a logistic distribution.
While similar to the normal distribution in shape, it exhibits slightly heavier tails but lighter tails than the Student's t-distribution.
If X follows a hyperbolic secant distribution, then exp(X) follows a half-Cauchy distribution.
The ratio of two independent standard normal random variables follows a scaled version of the hyperbolic secant distribution.

The HyperbolicSecantDistribution class

The hyperbolic secant distribution is implemented by the HyperbolicSecantDistribution class. It has one constructor that takes the 2 parameters in the following order: location and scale. The following constructs a hyperbolic secant distribution with location 2 and scale 3:

var hyperbolicSecant = new HyperbolicSecantDistribution(2.0, 3.0);

Visual Basic

Dim hyperbolicSecant = New HyperbolicSecantDistribution(2.0, 3.0)

Visual Basic

No code example is currently available or this language may not be supported.

let hyperbolicSecant = new HyperbolicSecantDistribution(2.0, 3.0)

Note that evaluation of the Cumulative Distribution Function (CDF) of a hyperbolic secant distribution is straightforward because it has a closed form. Evaluating the inverse CDF is also straightforward.

The HyperbolicSecantDistribution class has two specific properties that correspond to the parameters of the distribution. The LocationParameter and ScaleParameter properties return the location and scale parameters, respectively.

References

For more information on the hyperbolic secant distribution, refer to the following sources:

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 23. Wiley Series in Probability and Statistics.
"Hyperbolic Secant Distribution." Wikipedia, https://en.wikipedia.org/wiki/Hyperbolic_secant_distribution.