Parameter Estimation for Multi-state Coherent Series and Parallel Systems with Positively Quadrant Dependent Models
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Parameter Estimation for Multi-state Coherent Series and Parallel Systems with Positively Quadrant Dependent Models Leena Kulkarni and Sanjeev Sabnis IIT Bombay, Mumbai, India
Leena Kulkarni SVKM’s NMIMS (Deemed-to-be University), Mumbai, India
Sujit K. Ghosh North Carolina State University, Raleigh, USA Abstract 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. AMS (2000) subject classification. Primary 62, Secondary 62N05, 62F10, 62F12. Keywords and phrases. Multi-state series system, Generalized method of moments, Maximum likelihood estimation, Positively quadrant dependent, Farlie-Gumbel-Morgenstern distribution
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Introduction
Any coherent system consisting of a number of components having different levels of work efficiency can be considered as a multi-state coherent system (MSCS). The performance of a MSCS can be measured if the structure of the system is known. However, when the exact structure of the system is very complex, the determination of reliability of a MSCS can become extremely challenging. Excellent references that deal with the study of various theoretical as well as many practical applications of multi-state coherent systems are the well known books by Natvig (2010) & Lisnianski and Levitin (2003). Specifically, Natvig’s book discusses among other topics bounds for system availability and unavailability, Bayesian assessment of system availability, various measures of importance of system components along with a few examples, whereas that of Lisnianski and Levitin (2003) c
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