Robust estimation of dependent competing risk model under interval monitoring and determining optimal inspection intervals

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Robust estimation of dependent competing

risk model under interval monitoring and

determining optimal inspection intervals

Shuvashree Mondal∗1& Shanya Baghel2

Abstract

Recently, a growing amount interest is quite evident in modelling dependent competing risks

in life time prognosis problem. In this work, we propose to model the dependent competing

risks by Marshal-Olkin bivariate exponential distribution. The observable data consists of

number of failures due to diﬀerent causes across diﬀerent time intervals. The failure count

data is common in instances like one shot devices where state of the subjects are inspected at

diﬀerent inspection times rather than the exact failure times. The point estimation of the life

time distribution in presence of competing risk has been studied through divergence based

robust estimation method called minimum density power divergence estimation (MDPDE).

The testing of hypothesis is performed based on a Wald type test statistic. The inﬂuence

function is derived both for the point estimator and the test statistic, which reﬂects the de-

gree of robustness. Another, key contribution of this work is to determine the optimal set

of inspection times based on some predeﬁned objectives. This article presents determination

of multi criteria based optimal design. Population based heuristic algorithm non-dominated

sorting-based multiobjective Genetic algorithm is exploited to solve this optimization prob-

lem.

Key Words and Phrases: Divergence Based Robust Estimation, Competing Risk, Multi Ob-

jective Optimization, Marshal Olkin Bivariate Exponential Distribution, Inﬂuence function.

AMS Subject Classifications: 62F10, 62F03, 62H12.

1,2Department of Mathematics and Computing, Indian Institute of Technology (Indian School of

Mines) Dhanbad, Dhanbad- 826004, India.

∗Correspondence: Shuvashree Mondal, Department of Mathematics and Computing, Indian

Institute of Technology (Indian School of Mines), Dhanbad 826004, India.

Email: shuvasri29@iitism.ac.in

arXiv:2210.05911v1 [stat.AP] 12 Oct 2022

1 Introduction

Competing risk data arises when an event takes place due to diﬀerent simultaneously eﬀective

causes. The occurrence of an event due to one speciﬁc cause precludes one from observing the

occurrence of events due to the other causes. In the literature, a signiﬁcant amount of work has

been done in competing risk problem. Crowder [15] provided a monograph on the analysis of

diﬀerent competing risk models. Prentice et al. [33] analysed failure time data in the competing

risk environment. Austin et al. [1] proposed the analysis of survival data in the presence of

competing risks. Balakrishnan et al. [7, 8] studied estimation of diﬀerent lifetime distributions in

presence of competing risks. Balakrishnan et al. [9] provided Bayesian inference under competing

risk setup. Wang et al. [35] studied competing risk failure time data for a frailty-copula model.

Dutta and Kayal [20] conducted inferential study under censoring scheme on competing risk data.

Most of those articles present stochastically independent competing risks in action. Recently, a

growing interest is quite evident in modelling dependent competing risks in life time prognosis

problem. Justiﬁcation of such modelling lies in instances like shock model originally found in

Marshall and Olkin [30]. Suppose, in a system with two components, shock 1 is responsible for

failure of component 1, shock 2 is for component 2 while shock 3 results in failure of both the

components. In such case, the system fails if any one component fails, it is indeed an example of

dependent competing risk set up. In the literature, we ﬁnd the study on dependent competing risk

in Bai et al. [2], Cai et al. [13], Feizjavdian and Hashemi [23], Kundu [26], Kundu and Mondal

[25], Shen and Xu [34], Lyu et al. [29] and references therein.

In this work, we explore the study of statistical inference of the life time distribution under de-

pendent competing risk set up. It is assumed that life time under two dependent competing risks

follows a bivariate Marshall Olkin distribution. The subjects of interest are put on life testing

experiment which continues until a pre-speciﬁed time point. The observable data consists of num-

ber of failures due to diﬀerent causes across diﬀerent time intervals. The failure count data is

common in instances like one shot devices where state of the subjects are inspected at diﬀerent

inspection times rather than the exact failure times, readers may see Blakrishnan and Ling [4, 5],

Balakrishnan et al. [6] for references.

In inference study, conventional point estimation method is the maximum likelihood estimation

(MLE) which is quite popular because of its well-known properties such as asymptotic eﬃciency,

consistency, suﬃciency, invariance transformation. But in presence of outliers, MLE can not

perform well. Basu et al [10] proposed divergence based robust estimation method called minimum

density power divergence estimator (MDPDE) by incorporating a tuning parameter which brings

a trade-oﬀ between robustness and eﬃciency. In this work, along with the MLE, we develop the

MDPDE in dependent competing risk set up based on the failure count data.

Along with point estimation, testing of hypothesis is an essential component in inference study.

In this article, we present hypothesis testing based on the robust MDPDE. The null hypothesis is

constituted based on the equality of the scale parameters of the competing risks. In this regard

a Wald type test statistic is developed based on the asymptotic distribution of the MDPDE. An

approximation method is applied for the power calculation.

The robustness of any statistic can be assessed by its inﬂuence function. In the context of dependent

competing risk set-up, the inﬂuence functions are computed both for the MDPDE and the Wald

type test statistic. Through numerical experiment also, we depict the robustness of the MDPDE

compared to MLE.

Apart from inference study, in interval monitoring set-up, it is essential to set the inspection

times such that the experiment serves diﬀerent goals of the experimenter adequately. In this

context, we desire the precision of the estimator to be as high as possible along with minimum

budget for the experiment. We try to achieve both the goals through multi-objective optimization.

Population based heuristic algorithm, Genetic Algorithm (GA) is implemented which returns a set

of Pareto optimal solutions. In the literature, Genetic algorithm has been successfully implemented

in diﬀerent situations. Readers may refer to Faraz [22], Liu et al. [28], Parkinson [32], Yang et

al.[36]. In this work, we exploit a version of non-dominated sorting GA called NSGA-II proposed

by Deb et al. [16].

The rest of the article goes as follows. In Section 2, we put down the description of the model

along with the study of likelihood function and the maximum likelihood estimators. We derive the

robust density power divergence estimator in section 3. Section 4 provides the study of the testing

of hypothesis based on the robust estimator. In Section 5, we study the inﬂuence functions for

both point estimator and the test statistic. Determination of optimal inspection times is studied

in Section 6. In Section 7, an extensive numerical experiment along with a real data analysis for

illustration purposes are presented for the performance evaluation of the developed methods.

2 Model Description

In this section, we brieﬂy describe the Marshall-Olkin Bivariate Exponential (MOBE) distribution

as the life time model followed by description of the model layout.

The cumulative distribution function (cdf) of an exponential distribution with scale parameter λ

is deﬁned as

FExp(x) = 1 −e−λ x

and the pdf is derived as

fExp(x;α, λ) = λ e−λ x,where x > 0, λ > 0,

and it will be denoted by Exp(λ). Suppose, U0∼Exp(λ0), U1∼Exp(λ1), U2∼Exp(λ2) and they

are independently distributed. Deﬁne, X1=min{U0, U1}and X2=min{U0, U2}. The bivariate

random vector (X1, X2) is said to follow Marshall-Olkin bivariate Exponential distribution denoted

by MOBE(λ0, λ1, λ2).The joint survival function of (X1, X2) can be derived as,

SM OBE (x1, x2) = P(X1> x1, X2> x2) = P(U0> z, U1> x1, U2> x2)

=e−(λ0z+λ1x1+λ2x2)

where z=max{x1, x2}.Therefore, the joint probability density function (PDF) of (X1, X2) can

be obtained as

fM OBE (x1, x2) = 





λ1(λ0+λ2)e−λ1x1−(λ0+λ2)x20< x1< x2<∞

λ2(λ0+λ1)e−(λ0+λ1)x1−λ2)x20< x2< x1<∞

λ0e−λx,0< x1=x2=x < ∞.

(1)

where, λ=λ0+λ1+λ2.

Suppose n units are put on the life testing experiment and each unit is subject to two competing

risks. Let T1denote the failure time due to risk 1 and T2denote the same for risk 2. Here,

we assume that (T1, T2)∼MOBE(λ0, λ1, λ2).Under these competing risk set-up, the observable

failure time is T=min(T1, T2).In the life testing experiment, at diﬀerent inspection times say

τ1, . . . , τK,the experimenter will observe the number of failures in each interval due to the com-

peting causes and the experiment is terminated at τKtime point. Let Nibe the number of failures

which take place in (τi−1, τi] interval for i= 1, . . . , K where τ0= 0. Nican be decomposed as

Ni=Ni0+Ni1+Ni2, where Ni1(Ni2) is the number of failures due to cause l (cause 2) and Ni0is

the number of failure due to both the causes. Let Nsbe the censored units at the time point τK,

therefore, Ns=n−PK

i=1 P2

l=0 Nil.

It is evident that, (N11, N12, N10,··· , NK1, NK2, NK0, Ns)∼Multinomial (n,p),with the probabil-

ity vector p= (p11, p12, p10,··· , pK1, pK2, pK0, ps), where for i= 1, . . . , K,

pi1=P(τi−1< T1≤τi, T2> T1)

=Zτi

τi−1Z∞

fM OBE (x1, x2)I(x1< x2)dx1dx2

=λ1

λe−λτi−1−e−λτi

pi2=P(τi−1< T2≤τi, T1> T2)

=λ2

λe−λτi−1−e−λτi

pi0=P(τi−1< T1=T2≤τi)

=λ0

λe−λτi−1−e−λτi,and

ps=P(min(T1, T2)> τK)

=e−λτK.

Based on the failure count data across the intervals, the likelihood function can be written as

L(θ)∝ K

i=1

l=0

pNil

il !×pNs

=λPK

i=1 Ni1

1λPK

i=1 Ni2

2λPK

i=1 Ni0

λPK

i=1 Ni×

i=1 e−λτi−1−e−λτiNi×e−λNsτK

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RobustestimationofdependentcompetingriskmodelunderintervalmonitoringanddeterminingoptimalinspectionintervalsShuvashreeMondal1&ShanyaBaghel2AbstractRecently,agrowingamountinterestisquiteevidentinmodellingdependentcompetingrisksinlifetimeprognosisproblem.Inthiswork,weproposetomodelthedependentcompeti...

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Robust estimation of dependent competing risk model under interval monitoring and determining optimal inspection intervals

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