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."
Publié le : 1981-07-14
Classification:
Testing with replacement,
renewal testing,
product limit estimator,
Kaplan-Meier estimator,
censoring,
62G05,
60K10
@article{1176345525,
author = {Gill, R. D.},
title = {Testing with Replacement and the Product Limit Estimator},
journal = {Ann. Statist.},
volume = {9},
number = {1},
year = {1981},
pages = { 853-860},
language = {en},
url = {http://dml.mathdoc.fr/item/1176345525}
}
Gill, R. D. Testing with Replacement and the Product Limit Estimator. Ann. Statist., Tome 9 (1981) no. 1, pp. 853-860. http://gdmltest.u-ga.fr/item/1176345525/