## The Annals of Statistics

- Ann. Statist.
- Volume 15, Number 4 (1987), 1536-1547.

### Strong Consistency of a Nonparametric Estimator of the Survival Function with Doubly Censored Data

Myron N. Chang and Grace L. Yang

#### Abstract

A double censoring mechanism is such that each variable $X$ in the sample is observable if and only if $X$ is within the observation interval $\lbrack Z, Y \rbrack$. Otherwise, we can only determine whether $X$ is less than $Z$ or greater than $Y$ and observe $Z$ or $Y$ correspondingly. This kind of censoring occurs often in collecting lifetime data. Our problem is to estimate the survival function of $X, S_X(t) = P \lbrack X > t \rbrack$, from a doubly censored sample, where $X$ is assumed to be independent of the random interval $\lbrack Z, Y \rbrack$. We establish sufficient conditions for which $S_X(t)$ is identifiable and then prove the strong consistency of the self-consistent estimator $\hat{S}_X(t)$ for $S_X(t)$. This investigation generalizes the results available for the right censored data.

#### Article information

**Source**

Ann. Statist. Volume 15, Number 4 (1987), 1536-1547.

**Dates**

First available: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176350608

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aos/1176350608

**Mathematical Reviews number (MathSciNet)**

MR913572

**Zentralblatt MATH identifier**

0629.62040

**Subjects**

Primary: 62G05: Estimation

Secondary: 62G99: None of the above, but in this section

**Keywords**

Survival distributions doubly censored data self-consistent estimation uniform strong consistency

#### Citation

Chang, Myron N.; Yang, Grace L. Strong Consistency of a Nonparametric Estimator of the Survival Function with Doubly Censored Data. The Annals of Statistics 15 (1987), no. 4, 1536--1547. doi:10.1214/aos/1176350608. http://projecteuclid.org/euclid.aos/1176350608.