Asymptotically Minimax Estimators for Distributions with Increasing Failure Rate

Abstract

We construct nonparametric estimators of the distribution function F and its hazard function in the class of all increasing failure rate (IFR) distribution. Denoting the empirical distribution and empirical hazard function by $F_n$ and $H_n$ , respectively, let $C_n$ be the greatest convex minorant of $H_n$, and $G_n$ the distribution with hazard function $C_n$. The estimator $G_n$ is itself IFR. We prove that under suitable restrictions on F, and for any fixed $\lambda$ with $F(\lambda)<1$, $sup_{x\leq\lambda {n^{1/2}}}|C_n(x)-H_n(x)|$ and $sup_{x\leq\lambda{n^{1/2}}}|G_n(x)-F_n(x)|$ both tend to zero in probability. This means that $G_n$ and $F_n$ are asymptotically $n^{1/2}$ equivalent. It follows from Millar (1979) that $F_n$ is asymptoticallyminimax among the class of all IFR distributions for a large class of loss functions. This property extends to our estimator $G_n)$ under some restrictions.