Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 3 , pp. 416-430

ON THE STRONG UNIFORM CONSISTENCY OF THE PRODUCT LIMIT ESTIMATOR

By

YU QIQING, State University of New York at Binghamton

and

LI LINXIONG, University of New Orleans

SUMMARY. Kaplan and Meier (1958) proposed the product limit estimator (PLE) as a generalized MLE of an unknown survival function $S(t)$ when the data is subject to right censoring, with an unknown distribution function $G(t)$. In this note the largest interval on which the PLE is strong consistent is found for arbitrary survival and censoring distribution functions. In particular, letting $\tau$ be the supremum of $t$ such that both $S(t)$ and $1 - G(t)$ are positive, it is shown that, \ben && \left \{ \begin {array}{l} \dsp {\lim _{n \raro \ity}} \; \dsp {\sup _{t < \tau}}|\hat {S}_{PL}(t) - S(t)| = 0 \mbox { a.s.} \\ \dsp {\lim _{n \raro \ity}} \; \dsp {\sup _{t \ge \tau}}|\hat {S}_{PL}(t) - S(\tau -)| = 0 \mbox { a.s.} \ \end{array} \right. \qquad \mbox { if } S(\tau -) > S(\tau ) \mbox { and } G(\tau - ) = 1; \\ && \left \{ \begin {array}{l} \dsp {\lim _{n \raro \ity}} \; \dsp {\sup _{t \le \tau}}|\hat {S}_{PL}(t) - S(t)| = 0 \mbox { a.s.} \\ \dsp {\lim _{n \raro \ity}} \; \dsp {\sup _{t > \tau}}|\hat {S}_{PL}(t) - S(\tau )| = 0 \mbox { a.s.} \end{array} \right. \quad \qquad \mbox{otherwise.}.

 AMS (1980) subject classification. Primary 62F12; secondary 62A20.

Key words and phrases. Right censorship, uniform strong consistency, nonparametric estimation.

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This paper in Mathematical Reviews.