[Paper Review] Products of normal, beta and gamma random variables: Stein operators and distributional theory

Robert E. Gaunt|arXiv (Cornell University)|Jul 28, 2015

Random Matrices and Applications4 citations

TL;DR

This paper extends Stein's method to products of independent beta, gamma, generalized gamma, and mean-zero normal random variables by deriving novel Stein operators for these mixed products. It establishes closed-form expressions involving the Meijer G-function for the probability density and characteristic functions of such products, unifying classical Stein operators for normal, beta, and gamma distributions as special cases.

ABSTRACT

In this paper, we extend Stein's method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. In particular, we obtain Stein operators for mixed products of these distributions, which include the classical beta, gamma and normal Stein operators as special cases. These operators lead us to closed-form expressions involving the Meijer $G$-function for the probability density function and characteristic function of the mixed product of independent beta, gamma and central normal random variables.

Motivation & Objective

• To extend Stein's method beyond the normal distribution to products of independent beta, gamma, generalized gamma, and mean-zero normal random variables.
• To derive Stein operators for mixed products of these distributions, generalizing classical operators for individual distributions.
• To provide closed-form expressions for the probability density and characteristic functions of such products using the Meijer G-function.
• To unify and generalize existing Stein operators for normal, beta, and gamma distributions within a single framework.

Proposed method

• Derives a generalized Stein operator for products of independent beta, gamma, generalized gamma, and mean-zero normal random variables.
• Uses the Mellin transform and properties of the Meijer G-function to express the density and characteristic function of the product distribution.
• Establishes that the product of independent beta, gamma, and normal variables has a distribution expressible via the Meijer G-function.
• Applies known integral identities and asymptotic expansions of the Meijer G-function to analyze the resulting distributions.
• Connects the derived operators to existing Stein operators for the normal, beta, and gamma distributions as special cases.
• Employs differential equations satisfied by the Meijer G-function to validate the derived Stein operators.

Experimental results

Research questions

• RQ1 How can Stein's method be extended to handle products of independent beta, gamma, generalized gamma, and mean-zero normal random variables?
• RQ2 What is the form of the Stein operator for a mixed product of these distributions, and how does it generalize classical operators for individual distributions?
• RQ3 Can closed-form expressions for the probability density and characteristic functions of such products be derived using special functions?
• RQ4 What is the role of the Meijer G-function in characterizing the distribution of these products?
• RQ5 How do the derived operators relate to known Stein operators for the normal, beta, and gamma distributions?

Key findings

• The paper derives a new Stein operator for the product of independent beta, gamma, and mean-zero normal random variables, generalizing classical operators for individual distributions.
• The probability density function of the product is expressed in closed form using the Meijer G-function, specifically as a generalized hypergeometric function.
• The characteristic function of the product distribution is also shown to have a closed-form expression in terms of the Meijer G-function.
• The product of independent normal random variables follows a distribution expressible via the Meijer G-function, with a special case involving the modified Bessel function $ K_0 $.
• The derived operators reduce to the classical Stein operators for the normal, beta, and gamma distributions when specialized to single-component products.
• The framework allows for the derivation of bounds on probability distances via the solution of inhomogeneous differential equations, extending the applicability of Stein's method to product distributions.