Maximum likelihood inference about parameters of geometric lifetimes of heterogeneous components from data collected till failure of a k-out-of-n:G system

In this paper, we study parametric inference from data collected till the breakdown of a k-out-of-n:G system. Component lifetimes are assumed to be independent but not necessarily identically distributed geometric random variables. We find maximum likelihood estimators (MLE’s) of parameters of the geometric lifetimes and discuss their existence. We propose two interpretations of the situation when the MLE does not exist. Next, under the two interpretations, we find closed-form formulas for the expectation and mean squared error of the MLE’s when the system has a series structure. In the case of non-series systems, we study these quantities by Monte Carlo simulations. Applications of our results to Type-II right censored data coming from two or more populations and to lifetime data with multiple causes of failures are also indicated.