## The Annals of Statistics

- Ann. Statist.
- Volume 20, Number 2 (1992), 1022-1039.

### Large Sample Study of Empirical Distributions in a Random-Multiplicative Censoring Model

#### Abstract

Consider an incomplete data problem with the following specifications. There are three independent samples $(X_1, \ldots, X_m), (Z_1, \ldots, Z_n)$ and $(U_1, \ldots, U_n)$. The first two samples are drawn from a common lifetime distribution function $G$, while the third sample is drawn from the uniform distribution over the interval $(0,1)$. In this paper we derive the large sample properties of $\hat{G}_{m,n}$, the nonparametric maximum likelihood estimate of $G$ based on the observed data $X_1, \ldots, X_m$ and $Y_1, \ldots, Y_n$, where $Y_i \equiv Z_iU_i, i = 1, \ldots, n$. (The $Z$'s and $U$'s are unobservable.) In particular we show that if $m$ and $n$ approach infinity at a suitable rate, then $\sup_t|\hat{G}_{m,n}(t) - G(t)| \rightarrow 0$ (a.s.), $\sqrt{m + n}(\hat{G}_{m,n} - G)$ converges weakly to a Gaussian process and the estimate $\hat{G}_{m,n}$ is asymptotically efficient in a nonparametric sense.

#### Article information

**Source**

Ann. Statist., Volume 20, Number 2 (1992), 1022-1039.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176348668

**Digital Object Identifier**

doi:10.1214/aos/1176348668

**Mathematical Reviews number (MathSciNet)**

MR1165604

**Zentralblatt MATH identifier**

0761.62056

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62G20: Asymptotic properties

**Keywords**

Censored data informative censoring nonparametric maximum likelihood estimation weak convergence efficiency survival function

#### Citation

Vardi, Y.; Zhang, Cun-Hui. Large Sample Study of Empirical Distributions in a Random-Multiplicative Censoring Model. Ann. Statist. 20 (1992), no. 2, 1022--1039. doi:10.1214/aos/1176348668. https://projecteuclid.org/euclid.aos/1176348668