Properties and approximations of a Bessel distribution for data science applications

TL;DR

This paper investigates the properties of a Bessel distribution derived from Gaussian and chi-squared variables, offering approximations and closed-form expressions to facilitate its use in data science applications.

Contribution

It introduces novel approximations and properties of the Bessel distribution, including a Laplace approximation and a closed-form CDF, enhancing computational efficiency.

Findings

Laplace distribution approximates Bessel quantiles with <10% error

Closed-form CDF using Struve functions derived

Empirical power-series approximation tested successfully

Abstract

This paper presents properties and approximations of a random variable based on the zero-order modified Bessel function that results from the compounding of a zero-mean Gaussian with a $\chi^2_1$-distributed variance. This family of distributions is a special case of the McKay family of Bessel distributions and of a family of generalized Laplace distributions. It is found that the Bessel distribution can be approximated with a null-location Laplace distribution, which corresponds to the compounding of a zero-mean Gaussian with a $\chi^2_2$-distributed variance. Other useful properties and representations of the Bessel distribution are discussed, including a closed form for the cumulative distribution function that makes use of the modified Struve functions. Another approximation of the Bessel distribution that is based on an empirical power-series approximation is also presented. The approximations are tested with the application to the typical problem of statistical hypothesis testing. It is found that a Laplace distribution of suitable scale parameter can approximate quantiles of the Bessel distribution with better than 10% accuracy, with the computational advantage associated with the use of simple elementary functions instead of special functions. It is expected that the approximations proposed in this paper be useful for a variety of data science applications where analytic simplicity and computational efficiency are of paramount importance.