Article

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 56, Issue 1, pp 75-86

First online:

A LIL type result for the product limit estimator

  • A. FöldesAffiliated withMathematical Institute of the Hungarian Academy of Sciences
  • , L. RejtőAffiliated withMathematical Institute of the Hungarian Academy of Sciences

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Summary

Let X 1,X 2,...,X n be i.i.d. r.v.'-s with P(X>u)=F(u) and Y 1,Y 2,...,Y n be i.i.d. P(Y>u)=G(u) where both F and G are unknown continuous survival functions. For i=1,2,...,n set δ i=1 if X i ≦Y i and 0 if X i >y i , and Z i =min {itXi, Yi}. One way to estimate F from the observations (Z i ,δ i ) i=l,...,n is by means of the product limit (P.L.) estimator F n * (Kaplan-Meier, 1958 [6]).

In this paper it is shown that F n * is uniformly almost sure consistent with rate O(√log logn/√n), that is P(sup ¦F n * (u)− F(u)¦=0(√log log n/n)=1 −∞<u<+∞ if G(T F )>0, where T F =sup{xF(x)>0}.

A similar result is proved for the Bayesian estimator [9] of F. Moreover a sharpening of the exponential bound of [3] is given.