Exact Maximum Likelihood Estimation Using Masked System Data

This paper estimates component reliability from masked series-system life data, viz, data where the exact component causing system failure might be unknown. We extend the results of Usher & Hodgson (1988) by deriving exact maximum likelihood estimators (MLE) for the general case of a series system of 3 exponential components with independent masking. Their previous work shows that closed-form MLE are intractable, and they propose an iterative method for the solution of a system of 3 Nonlinear likelihood equations. They do not, however, prove convergence for their iterative method. As such, we show how this system of Nonlinear equations can be replaced by a single quartic equation, whose solution is straight-forward. Since it does not depend upon the convergence of numerical solution algorithms, the results are exact. Though the resulting estimators are somewhat lengthy & cumbersome to find manually, they can be written as a straightforward computer code. The calculations can then be easily performed on a personal computer. This method for reducing the likelihood equations to simpler-to-solve forms can be extended readily to a higher number of components. In many cases for more than 3 components it is easier while for others it is more complicated; even in the more complicated cases, this simplification makes the problems much more tractable.