*J. Japan Statist. Soc.,* Vol. 34 (No. 2), pp. 153-172, 2004

## Unbiased Estimation of Functionals under Random Censorship

Akio Suzukawa

**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 *F*_{n}, where *n* is sample size. A natural estimator of *∫* *φdF* is a KM integral, *∫* *φdF*_{n}. 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.

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