Sankhya: The Indian Journal of Statistics

2000, Volume 62, Series A, Pt. 1, pp. 80--85

THE JUMP SIZES OF THE PRODUCT-LIMIT ESTIMATOR UNDER RAMDOM CENSORSHIP

By

BIH-SHEUE SHIEH, Chia-Nan College of Pharmacy and Science, Taiwan

and

CHEUN-DER LEA ,National Cheng-Kung University, Taiwan

SUMMARY. In this study, we investigate the supremum of the jump sizes of the Efron's version of the product-limit (PL)-estimator $F_n$ and obtain the following result: If $P\{X_i\le Y_i\}>p,\>0i=1,\cdots,n$ then $\sup_i\Delta_i=o(n^{-p})$ where $\{\Delta_i\}$ denote the jump sizes of $F_n$. Using this result, we construct some quite smooth estimates $\widehat F_n$ such that $\sup_{x}|\widehat F_n(x)-F_n(x)|=o(n^{-p})$.

AMS (1991) subject classification. 62E20, 62G30.

Key words and phrases. Survival function, censored data, PL-estimator.

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