Maximum Likelihood Estimation of Life-Distributions from Renewal Testing

Abstract

In 1965 Marshall and Proschan (1965) (see also Grenander (1956)), considered the maximum likelihood estimation for life-distributions with monotone failure rate over the support of the distribution functions. They considered data arising from a testing plan which does not allow censoring, time-truncation or replacements. In the present paper we consider the maximum likelihood estimation of life-distributions with monotone failure rates over the interval $\lbrack 0, T)$, where $T$ is a fixed positive real number, and no other assumptions about the distribution or its failure rate are given outside that interval. The following renewal type testing plan is used, which allows for time-truncation and replacement. At time zero, the beginning of the testing, $n$ new items from the population to be tested are put on test. When an item fails it is instantaneously replaced with another new item from the same population and at time $T$ all testing is stopped. The maximum likelihood estimates of the distribution function and its failure rate over $\lbrack 0, T)$ are given and shown to be uniformly strongly consistent as $n$ tends to infinity.