TY - JOUR

T1 - Cumulative damage and times of occurrence for a multicomponent system

T2 - A discrete time approach

AU - Fierro, Raúl

AU - Leiva, Víctor

AU - Maidana, Jean Paul

N1 - Funding Information:
The authors thank the Editor-in-Chief, an Associate Editor, and two anonymous referees for their constructive comments on an earlier version of this manuscript, which resulted in this improved version. The research work of Raúl Fierro and Víctor Leiva was partially supported by the Chilean government under grant FONDECYT1160868. Jean Paul Maidana is the recipient of a doctoral grant FIB-UV from the Universidad de Valparaíso.
Funding Information:
The authors thank the Editor-in-Chief, an Associate Editor, and two anonymous referees for their constructive comments on an earlier version of this manuscript, which resulted in this improved version. The research work of Raúl Fierro and Víctor Leiva was partially supported by the Chilean government under grant FONDECYT 1160868 . Jean Paul Maidana is the recipient of a doctoral grant FIB-UV from the Universidad de Valparaíso.
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/11

Y1 - 2018/11

N2 - A discrete time stochastic model for a multicomponent system is presented, which consists of two random vectors representing a multivariate cumulative damage and their corresponding failure times. The times of occurrence of some events, for the system components, are correlated and their associate cumulative damages are assumed to be additive. Since, in general, it is not possible to obtain a closed form for the distribution of these random vectors, their asymptotic distribution is studied. A central limit theorem and a large deviation principle for the multivariate cumulative damage are derived. An application to neurophysiology is presented. Parameters associated with the mean and covariance matrix of the shocks are assumed known. Otherwise, they can be estimated through well-known methods. However, the critical levels (thresholds) of resistance for the components of the system are assumed to be unknown parameters. One of the objectives of this work is to carry out asymptotic statistical inference on these parameters. To this end, the asymptotic distribution of certain Mahalanobis type distances is studied, which enables us to estimate the parameters of interest and to test hypotheses concerning their values. Numerical results complete the analysis.

AB - A discrete time stochastic model for a multicomponent system is presented, which consists of two random vectors representing a multivariate cumulative damage and their corresponding failure times. The times of occurrence of some events, for the system components, are correlated and their associate cumulative damages are assumed to be additive. Since, in general, it is not possible to obtain a closed form for the distribution of these random vectors, their asymptotic distribution is studied. A central limit theorem and a large deviation principle for the multivariate cumulative damage are derived. An application to neurophysiology is presented. Parameters associated with the mean and covariance matrix of the shocks are assumed known. Otherwise, they can be estimated through well-known methods. However, the critical levels (thresholds) of resistance for the components of the system are assumed to be unknown parameters. One of the objectives of this work is to carry out asymptotic statistical inference on these parameters. To this end, the asymptotic distribution of certain Mahalanobis type distances is studied, which enables us to estimate the parameters of interest and to test hypotheses concerning their values. Numerical results complete the analysis.

KW - Asymptotic distribution

KW - Central limit theorem

KW - Hypothesis testing

KW - Large deviation principle

KW - Shock model

UR - http://www.scopus.com/inward/record.url?scp=85052731060&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2018.08.004

DO - 10.1016/j.jmva.2018.08.004

M3 - Article

AN - SCOPUS:85052731060

VL - 168

SP - 323

EP - 333

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

ER -