On the rate of uniform convergence of the product-limit estimator: Strong and weak laws
Let \(\{X_i\}\) and \(\{V_i\}\) be two independent sequences of nonnegative i.i.d. random variables with common distributions \(F\) and \(G\), respectively. In random censorship models, we observe \(\{Z_i, \delta_i,\;1\leq i\leq n\}\) with \(Z_i=X_i \wedge V_i\) and \(\delta_i= I_{\{X_i\leq V_i\}}\). Consider the (Kaplan-Meier) product-limit estimator of \(F\), \[ \widehat F_n(t)= 1-\prod_{0\leq s\leq t} \left(1-{dN_n(s) \over Y_n(s)} \right), \] where \(N_n(s)= \sum^n_{i=1} I_{\{Z_i\leq s,\;\delta_i=1\}}\), \(dN_n(s) =N_n(s) -N_n(s-)\) and \(Y_n(s)= \sum^n_{i=1} I_{\{Z_i\geq s\}}\). The subject of the paper is the important case \(F(\tau_H) <1\) and \(G(\tau_H) =1\), where \(\tau_H= \sup\{t: P(Z_1\leq t) <1\}\). An approximation of \(\widehat F_n\) by an average of i.i.d. random variables is used to derive necessary and sufficient conditions for the rate of (both strong and weak) uniform convergence of \(\widehat F_n\) on the whole line, that is for convergence \[ n^p \cdot \sup_{t\leq \tau_H} \bigl|\widehat F_n(t)- F(t)\bigr|\to 0, \quad 0<p <1/2 \] a.s. or in probability, respectively. These results fill a long standing gap in the asymptotic theory of survival analysis. The exact order \(p\) of the weak uniform convergence is determined, and a consistent estimator of \(p\) is proposed. A conjecture of \textit{R. D. Gill} [see: Lectures on survival analysis. Lect. Notes Math. 1581, 115-241 (1994; Zbl 0809.62028)] that the estimation of \(F(\tau_H)\) by \(\widehat F_n\) \((\tau_H- n^{-1/2})\) is accurate to order \(O_p ((\log n/n)^{1/2})\) is proved. This fact makes it possible to construct a confidence interval for \(F(\tau_H)\).
scientific article; zbMATH DE number 1053414