EM Algorithm and Positive Quadrant Dependence in the context of Multi-State Stress-Strength Model

Abstract

This research article is related to a multi-state stress-strength model proposed by Eryilmaz and İşçioğlu (2011). It deals with (i) demonstration of the use of EM algorithm for the estimation of parameters of distributions of random variables representing strength and two stress levels. The numerical results are illustrated using exponential distribution for each of the random variables under consideration, and (ii) comparison of results under the assumption of the random variables representing two stress levels being independent vis-à-vis they being positively quadrant dependent (PQD). The corresponding numerical results are based on Farlie-Gumbel- Morgenstern PQD distribution having non-identical exponential distributions as marginal distributions, and the distribution of strength variable is also an exponential distribution with a different parameter. As far as the EM algorithm related exercise is concerned, the numerical results, by and large, show that the EM algorithm, the one which makes clever use of data by pretending presence of missing observations, provides efficient estimators of the parameters than those provided by direct use of maximum likelihood (ML) estimators. For the latter exercise related to the PQD property, the numerical results highlight the fact that not only it is incorrect to assume two random variables representing two stress levels to be independent when in fact they are dependent, but the degree of dependence also can not be ignored.

Keywords

• Multi-State system
• EM algorithm
• PQD property
• Stress-strength model
• k-out-of-n system