Smooth Estimation of the Reliability Function

Authors: Kulasekera K.B.¹; Williams C.L.¹; Coffin M.²; Manatunga A.²

Source: Lifetime Data Analysis, Volume 7, Number 4, December 2001 , pp. 415-433(19)

Publisher: Springer

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Abstract:

Problems with censored data arise quite frequently in reliability applications. Estimation of the reliability function is usually of concern. Reliability function estimators proposed by Kaplan and Meier (1958), Breslow (1972), are generally used when dealing with censored data. These estimators have the known properties of being asymptotically unbiased, uniformly strongly consistent, and weakly convergent to the same Gaussian process, when properly normalized. We study the properties of the smoothed Kaplan-Meier estimator with a suitable kernel function in this paper. The smooth estimator is compared with the Kaplan-Meier and Breslow estimators for large sample sizes giving an exact expression for an appropriately normalized difference of the mean square error (MSE) of the two estimators. This quantifies the deficiency of the Kaplan-Meier estimator in comparison to the smoothed version. We also obtain a non-asymptotic bound on an expected lagran _1-type error under weak conditions. Some simulations are carried out to examine the performance of the suggested method.

Keywords: Kaplan-Meier estimator; Breslow estimator; censored data; Kernel smoothing; expected lagran -type error

Language: English

Document Type: Regular paper

Affiliations: 1: Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-1907, USA 2: Department of Biostatistics, Emory University, Atlanta, GA 30322, USA

Publication date: 2001-12-01