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

Anum Fatima, Gesine Reinert

TL;DR

CCR problems model failure times as the maximum or minimum of a random number $N$ of i.i.d. lifetimes, with $N$ independent of lifetimes. The authors develop Stein's method for the CCR class via a density-based score approach, enabling explicit distance bounds to simpler approximants such as the Poisson-exponential and exponential-geometric distributions. They derive general CCR comparison bounds and provide concrete results for PE, GPE, PG, EG, and EEG cases, alongside Lindeberg-type comparisons and asymptotic rates. The framework delivers quantitative convergence guarantees and practical tools for approximating complex CCR distributions in reliability, biostatistics, and risk modelling.

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.

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

TL;DR

CCR problems model failure times as the maximum or minimum of a random number of i.i.d. lifetimes, with independent of lifetimes. The authors develop Stein's method for the CCR class via a density-based score approach, enabling explicit distance bounds to simpler approximants such as the Poisson-exponential and exponential-geometric distributions. They derive general CCR comparison bounds and provide concrete results for PE, GPE, PG, EG, and EEG cases, alongside Lindeberg-type comparisons and asymptotic rates. The framework delivers quantitative convergence guarantees and practical tools for approximating complex CCR distributions in reliability, biostatistics, and risk modelling.

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.
Paper Structure (15 sections, 16 theorems, 140 equations, 1 table)

This paper contains 15 sections, 16 theorems, 140 equations, 1 table.

Table of Contents

  1. Introduction
  2. Stein's method for CCR distributions
  3. Stein's method for distributional comparisons
  4. CCR distributions
  5. Stein's method for the CCR class of distributions
  6. A general comparison approach
  7. Application to the Poisson-exponential distribution
  8. Approximating the distribution of the maximum of a random number of i.i.d. random variables by a PE distribution
  9. Approximating the generalised Poisson-exponential distribution
  10. Approximating the Poisson-geometric distribution
  11. Application to the exponential-geometric distribution
  12. The minimum of a geometric number of i.i.d. random variables
  13. Approximating the extended exponential-geometric distribution
  14. The maximum waiting time of sequence patterns in Bernoulli trials
  15. Further proofs

Key Result

Proposition 3.1

Let $W_{\alpha_1}(N, {\bf Y})$ and $W_{\alpha_2}(M, {\bf Z})$, for $\alpha_1, \alpha_2 \in \{-1, 1\}$ be CCR random variables with pdf's $f_Y$ and $f_Z$ and score functions $\rho_Y$ and $\rho_Z$. Then for any test function $h$ such that the $W_{\alpha_1}(N, {\bf Y})$-Stein equation eq:steingen for $ where $W = W_{\alpha_{2}}(M, {\bf Z})$ and $U_{\cdot}^{\alpha}$ is as in U_minmax. When the $Y_i$'s

Theorems & Definitions (41)

  • Example 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.2
  • ...and 31 more