The Annals of Statistics

Testing with Replacement and the Product Limit Estimator

R. D. Gill

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."

Article information

Source
Ann. Statist., Volume 9, Number 4 (1981), 853-860.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345525

Digital Object Identifier
doi:10.1214/aos/1176345525

Mathematical Reviews number (MathSciNet)
MR619288

Zentralblatt MATH identifier
0468.62039

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 60K10: Applications (reliability, demand theory, etc.)

Citation

Gill, R. D. Testing with Replacement and the Product Limit Estimator. Ann. Statist. 9 (1981), no. 4, 853--860. doi:10.1214/aos/1176345525. https://projecteuclid.org/euclid.aos/1176345525