Parameter Estimation for Multi-state Coherent Series and Parallel Systems with Positively Quadrant Dependent Models

A multi-state coherent system consists of multiple components each of which passes through a sequence of states and it is of interest to estimate the distribution of time spent by the components in different states. Although not practical, for mathematical convenience, it is usually assumed that the times spent by the components in various states are independent of each other. This paper considers three-state series and parallel systems and is based on the assumption that times spent by the components in various states are positively quadrant dependent (PQD) and the corresponding dependence is modeled using a Farlie Gumbel Morgenstern (FGM) distribution. To begin with it is shown that even when marginal distributions are assumed exponential, the resulting likelihood function leads to a complicated expression making maximum likelihood (ML) based inference computationally challenging. A generalized method of moment (GMM) estimation is shown to be relatively simpler not only computationally but also the method works for arbitrary marginal distributions. The estimates obtained by GMM are shown to be uniformly consistent under some mild regularity conditions. Finite sample performances of the ML and GMM are illustrated using FGM distribution with various parametric marginal distributions. In case of exponential marginals, it is shown that GMM compares favorably to ML although the former method does not require parametric assumption for the marginals. The proposed methods are also illustrated using a real case study data of a rare type of head and neck cancer.