## The Annals of Statistics

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

### 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."

#### 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.)

**Keywords**

Testing with replacement renewal testing product limit estimator Kaplan-Meier estimator censoring

#### 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