Geometric sums, size biasing and zero biasing

TL;DR

This paper characterizes gamma distribution via size biasing and zero biasing equivalence and applies Stein's method to derive bounds for gamma approximation of sums of nonnegative independent variables, including negative binomial sums.

Contribution

It introduces a novel characterization of gamma distribution through size and zero biasing and applies Stein's method for gamma approximation bounds in various sum distributions.

Findings

Gamma distribution characterized by size biasing equals zero biasing.

Derived bounds for gamma approximation of sums of independent nonnegative variables.

Extended gamma approximation techniques to negative binomial sums.

Abstract

The geometric sum plays a significant role in risk theory and reliability theory \cite{Kala97} and a prototypical example of the geometric sum is R\'enyi's theorem~\cite{Renyi56} saying a sequence of suitably parameterised geometric sums converges to the exponential distribution. There is extensive study of the accuracy of exponential distribution approximation to the geometric sum \cite{Sugakova95,Kala97,PekozRollin11} but there is little study on its natural counterpart of gamma distribution approximation to negative binomial sums. In this note, we show that a nonnegative random variable follows a gamma distribution if and only if its size biasing equals its zero biasing. We combine this characterisation with Stein's method to establish simple bounds for gamma distribution approximation to the sum of nonnegative independent random variables, a class of compound Poisson distributions and the negative binomial sum of random variables.