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September, 1986 Asymptotically Minimax Estimators for Distributions with Increasing Failure Rate
Jane-Ling Wang
Ann. Statist. 14(3): 1113-1131 (September, 1986). DOI: 10.1214/aos/1176350053


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.


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Jane-Ling Wang. "Asymptotically Minimax Estimators for Distributions with Increasing Failure Rate." Ann. Statist. 14 (3) 1113 - 1131, September, 1986.


Published: September, 1986
First available in Project Euclid: 12 April 2007

zbMATH: 0607.62042
MathSciNet: MR856809
Digital Object Identifier: 10.1214/aos/1176350053

Primary: 62G05
Secondary: 62G20, 62N05

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 3 • September, 1986
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