Estimation for Distributions with Monotone Failure Rate

Abstract

In this paper, we shall investigate a problem analogous to the one treated in Prakasa Rao [6]. We shall now suppose that there is a sample of $n$ independent observations from a distribution $F$ with monotone failure rate $r$ and the problem is to obtain a maximum likelihood estimator (MLE) of $r$ and its asymptotic distribution. Grenander [4] and Marshall and Proschan [5] have obtained the MLE of $r$ and the latter showed that these estimators are consistent. We obtain the asymptotic distribution of this estimator. The estimation problem is reduced at first to that of a stochastic process and the asymptotic distribution of MLE is obtained by means of theorems on convergence of distributions of stochastic processes. Methods used in this paper are similar to those in Prakasa Rao [6] and therefore, proofs are given only at places where they differ from the proofs of that paper. We shall consider the case of distributions with increasing failure rate (IFR) in detail. Results in the case of distributions with decreasing failure rate (DFR) are analogous to those of IFR and we shall state the main result in Section 7. Sections 2 and 3 deal with the definition and properties of distributions with monotone failure rate $r$. Some results related to the asymptotic properties of the MLE of $r$ are given in Section 4. The problem is reduced to that of a stochastic process in Section 5. The asymptotic distribution of the MLE is obtained in Section 6.