Testing with Replacement and the Product Limit Estimator

Abstract

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. nonnegative rv's with nondegenerate df $F$. Define $\tilde{N}(t) = {\tt\#}\{j: X_1 + \cdots + X_j \leq t\}$. In "testing with replacement" (also known as "renewal testing") $n$ independent copies of $\tilde{N}$ are observed each over the time interval $\lbrack 0, \tau \rbrack$ and we are interested in nonparametric estimation of $F$ based on these observations. We prove consistency of the product limit estimator as $n\rightarrow\infty$ for arbitrary $F$, and weak convergence in the case of integer valued $X_i$. We state the analogue of this result for continuous $F$ and briefly discuss the similarity of our results with those for the product limit estimator in the model of "random censorship."