J. Japan Statist. Soc., Vol. 34 (No. 2), pp. 153-172, 2004
Abstract. This paper is intended as an investigation of estimating functionals of a lifetime distribution F under right censorship. Functionals given by ∫ φdF, where φ's are known F-integrable functions, are considered. The nonparametric maximum likelihood estimator of F is given by the Kaplan-Meier (KM) estimator Fn, where n is sample size. A natural estimator of ∫ φdF is a KM integral, ∫ φdFn. However, it is known that KM integrals have serious biases for unbounded φ's. A representation of the KM integral in terms of the KM estimator of a censoring distribution is obtained. The representation may be useful not only to calculate the KM integral but also to characterize the KM integral from a point view of the censoring distribution and the biasedness. A class of unbiased estimators under the condition that the censoring distribution is known is considered, and the estimators are compared.
Key words and phrases: Censored data, Kaplan-Meier estimator, mean lifetime, product-limit estimator, survival data.