< List of probability distributions

Blog Title: Understanding the Half-Cauchy Distribution

One probability distribution that has gained attention in recent years is the **half-Cauchy distribution**, a heavy-tailed continuous probability distribution; it is a Cauchy distribution with a domain restricted to positive values greater than or equal to the location of its peak. In this article, we will explore this probability distribution in detail and understand its properties.

## What is the Half-Cauchy distribution?

A continuous probability distribution, the half-Cauchy distribution is derived by folding the standard Cauchy distribution on the origin, resulting in a version with only positive values. The Cauchy distribution has heavy tails, which means the probability of extreme events is higher compared to other distributions. The half-Cauchy inherits this property from the parent distribution, with probability density that extends to infinity.

One of the essential properties of the half-Cauchy distribution is its scale parameter, which determines the spread of the distribution. Lower values of the scale parameter result in a more concentrated distribution, while higher values generate flatter distributions. The half-Cauchy distribution’s median is zero, while the mean and variance are undefined since the distribution extends to infinity.

A continuous random variable *X* has a half-Cauchy distribution if its probability density function (PDF) is [1]

The cumulative distribution function (CDF) is

The half-Cauchy distribution is self-decomposable [2] (which means that it has the same distribution as the sum of a scaled down version of itself and an independent residual random variable) [3]. The half-Cauchy distribution is also infinity divisible [4].

Note that although the half-Cauchy is usually defined in terms of positive values (i.e., the right-hand half of the Cauchy distribution), In some rare cases, the left half of the Cauchy might be the distribution of interest, in which case this is usually specified.

## Half-Cauchy distribution use and implementation

In Bayesian statistics, the half-Cauchy distribution is often used as a prior distribution for the standard deviation parameter of a normal distribution because of its heavy tails. The heavy tails of the half-Cauchy distribution can handle outliers efficiently, making it suitable for models that require robustness against outliers.

The implementation of the half-Cauchy distribution is available in various statistical software such as R and Python. In R, the distribution is available in the {brms} package, which allows the user to estimate the distribution’s parameters using Bayesian methods. The distribution is available in Python’s {scipy.stats} module and is easily implemented through a few lines of code.

The half-Cauchy distribution has significant use cases in multiple disciplines, including machine learning and finance. In machine learning, the half-Cauchy distribution is used to specify prior distributions of hyperparameters in Bayesian neural networks. Additionally, in finance, the distribution is used to model skewness in financial returns.

## Relationship to other distributions

- The Half-Cauchy is a truncated Cauchy distribution where values at the peak or to the peak’s right have nonzero probability density.
- It is a special case of the Half-Student-t distribution.

In conclusion, the half-Cauchy distribution is a valuable tool in probability theory with many real-world applications. Its heavy tails make it optimal for modeling extreme events efficiently, which is crucial in many fields. With its availability in various software packages, the distribution is easily accessible for researchers and data scientists.

## References

[1] Jacob, E. & Jayakumar, K. On Half-Cauchy Distribution and Process. International Journal of Statistika and Mathematika, ISSN: 2277- 2790 E-ISSN: 2249-8605, Volume 3, Issue 2, 2012 pp 77-81

[2] A. Diedhiou. On the self-decomposability of the half-Cauchy distribution. J. MAth. Anal. Appl. 220(1), 42-64. 1998.

[3] Carr, P. et. al. (2007). Self-Decomposability and Option Pricing. Mathematical Finance, Vol. 17, No. 1 (January 2007), 31–57

[4] L. Bondesson, on the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative line. Scand. Actua. J. 1987(3), 225-247 (1987).