Statistical inference for intensity-based load sharing models with damage accumulation

Abstract

Consider a system in which a load exerted on it is equally shared between its components. Whenever one component fails, the total load is redistributed across the surviving components. This in turn increases the individual load applied to each of these components and therefore their risk of failure. Such a system is called a load sharing system.

In a load sharing system, the failure rate of a surviving component grows with the number of failed components. However, the risk of failure is likely to also depend on how long the surviving components were exposed to the shared load. This accumulation of damage within the system causes a continuous increase in the failure rate between consecutive component failures.

This thesis deals with the statistical inference for load sharing systems with damage accumulation that can be modelled in terms of its component failure rate. We identify the component failure rate as the stochastic intensity of a counting process, for which a parametric model can be specified - an intensity-based load sharing model with damage accumulation.

The first method of inference is the minimum distance estimator introduced by Kopperschmidt and Stute. They claim the strong consistency and asymptotic normality of this estimator, but we demonstrate that their proof of the asymptotic distribution is flawed. Our first important contribution is a corrected proof under slightly adjusted requirements.

The second method of inference is based on the K-sign depth test, a powerful and robust generalization of the classical sign test that was up to now mostly used with the residuals of a linear model. We present a procedure to obtain a "residual" counterpart in an intensity-based model via the hazard transformation of a point process. Moreover, we derive conditions on the model under which the 3-sign depth test is consistent.

The thesis closes by comparing these two methods with the established likelihood approach. To this end, we verify the applicability of the competing methods to the Basquin load sharing model with multiplicative damage accumulation recently proposed by Müller and Meyer. In a final simulation study, we assess the robustness of the methods in the presence of contaminated data. This study confirms that, in contrast to the other two approaches, the 3-sign depth test offers both a powerful and robust tool of statistical inference for intensity-based load sharing models with damage accumulation.