An equivalence based approach to dealing with uncertainty in maximum likelihood estimation

Sometimes it is difficult to obtain maximum likelihood estimates (MLE) directly from available data in presence of uncertainty. In this paper we develop an approach to dealing with uncertainty by introducing equivalent quantities of unknown variables from which concerned interests can be estimated directly. From expectation-maximization (EM) theory we treat the unknown variables as random variables and define their equivalent quantities as their mathematical expectations. To illustrate the concept of equivalence and its applications, we solve a specific MLE problem: To estimate component reliability from uncertain system life data. Two types of uncertainty for components are taken into account: status masking and left censored lifetime. We define and obtain equivalent failure and equivalent test time for components under different uncertain conditions. An EM algorithm based on the derived equivalent quantities is formulated to estimate component reliability from uncertain system life data. Compared to other methods, our approach based on equivalent quantities can handle more complex uncertainty. In addition, the convergence of the iterative estimation processes has been mathematically proved.