[Paper Review] Competing risks within shock models

Antonio Di Crescenzo, Maria Longobardi|ArXiv.org|Sep 1, 2008

Probability and Risk Models30 references3 citations

TL;DR

This paper introduces a competing risks framework within bivariate Poisson shock models, analyzing system failure due to one of two mutually exclusive shock types. It derives exact expressions for failure densities, survival functions, and conditional failure moments under three schemes: failure at the sum, minimum, or maximum of shocks reaching a random threshold M, with key results showing independence between failure time and cause in the sum-scheme and explicit formulas involving modified Bessel functions and cumulative Poisson distributions.

ABSTRACT

We consider a competing risks model, in which system failures are due to one out of two mutually exclusive causes, formulated within the framework of shock models driven by bivariate Poisson process. We obtain the failure densities and the survival functions as well as other related quantities under three different schemes. Namely, system failures are assumed to occur at the first instant in which a random constant threshold is reached by (a) the sum of received shocks, (b) the minimum of shocks, (c) the maximum of shocks.

Motivation & Objective

• To integrate competing risks into shock models by modeling system failure due to one of two mutually exclusive shock types.
• To analyze failure mechanisms under three distinct threshold schemes: sum, minimum, and maximum of shock counts.
• To derive exact expressions for failure densities, survival functions, and conditional failure moments given the cause of failure.
• To explore the stochastic dependence between failure time and cause of failure, particularly in the sum-scheme where they are shown to be independent.
• To extend the model to non-exclusive shock types using a bivariate Poisson process with shared components.

Proposed method

• Modeling shocks via two independent homogeneous Poisson processes $N_1(t)$ and $N_2(t)$ with rates $\lambda_1$ and $\lambda_2$.
• Defining failure sets $S_k$ based on the sum, minimum, or maximum of shock counts reaching a random threshold $M$ taking values in $\{1,2,\dots\}$.
• Using hazard rate decomposition and recursive expressions to derive subdensities $f_1(t)$ and $f_2(t)$ for failure due to each shock type.
• Employing the auxiliary function $E_k(x) = \sum_{j=0}^{k-1} \frac{x^j}{j!}$ and modified Bessel functions $I_j(\cdot)$ to express survival and density functions.
• Deriving the survival function $\overline{F}_T(t)$ via summation over failure sets and using the complementary cumulative distribution $\overline{P}_k$ of the threshold $M$.
• Extending the model to non-exclusive shocks using a bivariate Poisson process with three components: $\lambda_1$, $\lambda_2$, and $\lambda_3$ for shared shocks.

Experimental results

Research questions

• RQ1 Under what conditions does the failure time $T$ become independent of the cause $\delta$ of failure in a shock model with competing risks?
• RQ2 How do the failure densities and survival functions behave when failure is triggered by the sum of shocks reaching a random threshold $M$?
• RQ3 What are the exact mathematical expressions for the failure subdensities and survival function when failure occurs at the minimum or maximum of two shock counts?
• RQ4 How do the failure mechanisms differ when the threshold is based on the sum, minimum, or maximum of shock counts in a bivariate Poisson framework?
• RQ5 Can the model be extended to allow for non-exclusive shock types, and how would the failure set and hazard structure change in such a case?

Key findings

• In the sum-scheme, failure time $T$ and cause $\delta$ are stochastically independent, a non-trivial result derived from the structure of the hazard rates and subdensities.
• The survival function for the sum-scheme is given by $\overline{F}_T(t) = e^{-(\lambda_1+\lambda_2)t} \sum_{k=0}^{\infty} \overline{P}_k \left( \frac{(\lambda_1 t)^k}{k!} + \frac{(\lambda_2 t)^k}{k!} \right)$, showing a direct sum of Poisson-type terms.
• For the minimum-scheme, the failure subdensities are $f_1(t) = \lambda_1 e^{-(\lambda_1+\lambda_2)t} \sum_{k=1}^{\infty} p_k \frac{(\lambda_1 t)^{k-1}}{(k-1)!} E_k(\lambda_2 t)$, involving the cumulative Poisson function $E_k(x)$.
• In the maximum-scheme, the survival function is $\overline{F}_T(t) = e^{-(\lambda_1+\lambda_2)t} \sum_{k=0}^{\infty} \overline{P}_k \left( \frac{(\lambda_2 t)^k}{k!} E_k(\lambda_1 t) + \frac{(\lambda_1 t)^k}{k!} E_{k+1}(\lambda_2 t) \right)$, with $E_k(x)$ as defined.
• The sum of the subdensities in the minimum and maximum schemes reduces to $\lambda_i e^{-\lambda_i t} \sum_{k=1}^{\infty} p_k \frac{(\lambda_i t)^{k-1}}{(k-1)!}$, highlighting structural similarities.
• An approximation for the survival function in the sum-scheme with $h$ shock types is provided, generalizing the bivariate case to higher dimensions.