## The Annals of Statistics

- Ann. Statist.
- Volume 8, Number 5 (1980), 1002-1016.

### Large Sample Theory for an Estimator of the Mean Survival Time from Censored Samples

#### Abstract

This paper introduces and studies the large sample properties of an estimator for the mean survival time from censored samples. Let $X_1, \cdots, X_n$ be independent identically distributed random variables with $F(x) = P\lbrack X_1 > x \rbrack.$ Let $Y_1, \cdots, Y_n$ be independent identically distributed (and independent of $X_1, \cdots, X_n$) censoring times with $G(y) = P\lbrack Y_1 > y\rbrack.$ Based on observing only $Z_i = \min(X_i, Y_i)$ and which observations are censored (i.e., $X_i > Y_i$), we give a class of estimators of the mean survival time $\mu = \int^\infty_0F(x) dx.$ The estimators are of the form $\hat{\mu} = \int^{M_n}_0 \hat{F}(x)dx,$ where $M_n \uparrow \infty$ as $n \uparrow \infty$ and $\hat{F}$ is an estimator of $F$ depending on the $Z_i$'s and the censoring pattern. Conditions of $F, G$ and $\{M_n\}$ for the asymptotic normality of $\hat{\mu}$ are stated and proved in Section 2 based on approximations detailed in Section 3. Section 4 gives conditions for strong consistency of $\hat{\mu}$ with rates, while Section 5 examines the meaning of the conditions for the case of the negative exponential distributions for $F$ and $G.$

#### Article information

**Source**

Ann. Statist., Volume 8, Number 5 (1980), 1002-1016.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345138

**Digital Object Identifier**

doi:10.1214/aos/1176345138

**Mathematical Reviews number (MathSciNet)**

MR585699

**Zentralblatt MATH identifier**

0455.62030

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

Mean survival time estimation randomly right censored samples strong consistency asymptotic normality survival analysis and $U$-statistics

#### Citation

Susarla, V.; Ryzin, J. Van. Large Sample Theory for an Estimator of the Mean Survival Time from Censored Samples. Ann. Statist. 8 (1980), no. 5, 1002--1016. doi:10.1214/aos/1176345138. https://projecteuclid.org/euclid.aos/1176345138