# Multivariate Degradation Model with dependence due to Shocks

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1 LMAP - Laboratoire de Mathématiques et de leurs Applications Pau 2 ICD - Institut Charles Delaunay 3 UTT - Université de Technologie Troyes Troyes 4 ICD UTT - Université de Technologie Troyes Troyes, ICD - Institut Charles Delaunay

Abstract : The aim of this paper is to model degradation phenomena in a multi-unit context taking into account stochastic dependence between the units. Here, we intend to propose a model structure that allows in the same time enough complexity for the representation of phenomena and enough simplicity for the analytical calculations. More precisely, a multi-unit system is considered, which is submitted to a random stressing environment which arrives by shocks. The model takes into account two types of dependence between the components: firstly, a shock impacts all components at the same time; secondly, for a given shock, the deterioration increments of the different components are considered to be correlated. The intrinsic deterioration of the n say units is modeled through independent stochastic processes $Z t^{i} {t \geq 0}$, with $1 \leq i \leq n$. Given the usual nature of the degradation phenomena, is it seems reasonable to suppose that each $Z t^{i} {t \geq 0}$ should be a monotone process with continuous state space. Accordingly, the shocks are classically assumed to arrive independently, according to a Poisson process $N t {t \geq 0}$. The parameter estimation moment method and the reliability assessment are presented for any multi-unit systems with coherent structure. At last a numerical results is presented with a 3 units system.

Autor: Sophie Mercier - Anne Barros - Antoine Grall -

Fuente: https://hal.archives-ouvertes.fr/

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