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Abstract
This paper deals with parametric inference about the independent and identically distributed discrete lifetimes of components of a k-out-of-n system. We consider the maximum likelihood estimation assuming that the available data consists of component failure times observed up to and including the moment of the breakdown of the system. First, we provide general conditions for the almost sure existence of a strongly consistent sequence of maximum likelihood estimators (MLE’s). Then, we focus on three typical discrete failure distributions—the Poisson, binomial and negative binomial distributions—and prove that in these cases the MLE’s are unique, provided they exist, and that they are strongly consistent. Finally, we complete our results by Monte Carlo simulation study. Interestingly, the inference considered in the paper can be viewed as equivalent to one based on Type-II right censored discrete data. Therefore, our results can as well be applied to the case when Type-II right censored sample from a discrete distribution is observed.
Highlights
An important class of systems studied in reliability theory is one containing so called k-out-of-n systems
While reliability properties of k-out-of-n systems consisting of components with discrete lifetimes have been studied over the years; see Weiss (1962), Young (1970), Tank and Eryilmaz (2015), Dembinska (2018), Dembinska and Goroncy (2020) and Dembinska et al (2019), to the best of our knowledge results concerning inference about discrete lifetimes of components based on failure times in k-out-of-n systems are not known
We focus on maximum likelihood estimation of an unknown parameter of discrete distribution of component lifetimes of a k-out-of-n system
Summary
An important class of systems studied in reliability theory is one containing so called k-out-of-n systems. As technical structures having some redundancy, they find various applications in engineering when highly reliable products are needed, for example, they are used in design of servers in internet service or in design of automotive and aeronautic engines They have attracted substantial interest—a vast literature on k-out-of-n systems is available. While reliability properties of k-out-of-n systems consisting of components with discrete lifetimes have been studied over the years; see Weiss (1962), Young (1970), Tank and Eryilmaz (2015), Dembinska (2018), Dembinska and Goroncy (2020) and Dembinska et al (2019), to the best of our knowledge results concerning inference about discrete lifetimes of components based on failure times in k-out-of-n systems are not known.
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