Stein's method for distributions modelling competing and complementary risk problems

TL;DR

This paper develops Stein's method to compare and bound differences between complex CCR distributions, which model competing and complementary risks, using simpler distributions like Poisson-exponential and exponential geometric.

Contribution

It introduces a Stein's method framework specifically for CCR distributions, enabling effective approximation and comparison with simpler, tractable distributions.

Findings

Stein's method provides bounds for CCR distribution approximations.

Comparison with Lindeberg approach highlights advantages of Stein's method.

Explicit bounds are derived for Poisson-exponential and exponential geometric cases.

Abstract

Competing and Complementary risk (CCR) problems are often modelled using a class of distributions of the maximum, or minimum, of a random number of i.i.d. random variables; we call this class the CCR class of distributions. While the CCR distributions generally do not have an easy-to-calculate density or probability mass function, two special cases, namely the Poisson-exponential and the exponential geometric distributions, can easily be calculated. Hence, it is of interest to approximate CCR distributions with these simpler distributions. In this paper, we develop Stein's method for the CCR class of distributions to provide a general comparison approach to bound the distance between two CCR distributions and contrast this approach to bounds obtained using a Lindeberg argument. We detail the comparison for Poisson-exponential and exponential-geometric distributions.