Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 56, Issue 1, pp 75–86

A LIL type result for the product limit estimator

  • A. Földes
  • L. Rejtő
Article

DOI: 10.1007/BF00531975

Cite this article as:
Földes, A. & Rejtő, L. Z. Wahrscheinlichkeitstheorie verw. Gebiete (1981) 56: 75. doi:10.1007/BF00531975

Summary

Let X1,X2,...,Xn be i.i.d. r.v.'-s with P(X>u)=F(u) and Y1,Y2,...,Yn 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 δi=1 if Xi≦Yi and 0 if Xi>yi, and Zi=min {itXi, Yi}. One way to estimate F from the observations (Zi,δi) i=l,...,n is by means of the product limit (P.L.) estimator Fn* (Kaplan-Meier, 1958 [6]).

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

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • A. Földes
    • 1
  • L. Rejtő
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary