A LIL type result for the product limit estimator

SummaryLet 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 −∞0, where TF=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.