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

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

Random Matrices and Applications20 references6 citations

TL;DR

This paper extends Stein's method to products of independent beta, gamma, and standard normal random variables, deriving explicit Stein operators for mixed products. It establishes closed-form expressions using the Meijer G-function for the density and characteristic functions of these products, generalizing classical Stein operators for individual distributions.

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 to handle products of independent beta, gamma, and standard normal random variables.
• To derive explicit Stein operators for mixed products involving beta, gamma, and central normal distributions.
• To provide closed-form expressions for the probability density and characteristic functions of such products.
• To generalize classical Stein operators for beta, gamma, and normal distributions as special cases of the derived operators.
• To establish a distributional theory for products of these independent random variables using special functions.

Proposed method

• Derives Stein operators for products of independent beta, gamma, and standard normal random variables using distributional properties and differential operators.
• Applies the theory of special functions, particularly the Meijer G-function, to express the probability density function of the product distribution in closed form.
• Uses characteristic function representations based on the Meijer G-function to characterize the distribution of the product.
• Establishes that classical Stein operators for individual beta, gamma, and normal distributions emerge as special cases of the derived mixed-product operators.
• Employs integral representations and transformation techniques to link the product distribution to known special functions.
• Validates the framework by showing consistency with known results in the limit cases of single distributions.

Experimental results

Research questions

• RQ1 How can Stein's method be extended to products of independent beta, gamma, and standard normal random variables?
• RQ2 What are the explicit forms of the Stein operators for mixed products of these distributions?
• RQ3 Can closed-form expressions for the density and characteristic functions of such products be derived?
• RQ4 How do the classical Stein operators for beta, gamma, and normal distributions emerge as special cases of the generalized operators?
• RQ5 What role does the Meijer G-function play in characterizing the distribution of the product?

Key findings

• The paper derives explicit Stein operators for mixed products of independent beta, gamma, and standard normal random variables.
• Closed-form expressions for the probability density function of the product distribution are expressed using the Meijer G-function.
• The characteristic function of the product distribution is also represented in terms of the Meijer G-function.
• Classical Stein operators for beta, gamma, and normal distributions are recovered as special cases of the generalized operators.
• The framework provides a unified approach to distributional characterization of products involving these three families of distributions.
• The use of the Meijer G-function enables exact analytical representation of the product distribution's density and characteristic function.