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{x∶ F(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.
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Földes, A., Rejtő, L. A LIL type result for the product limit estimator. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56, 75–86 (1981). https://doi.org/10.1007/BF00531975
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DOI: https://doi.org/10.1007/BF00531975