## The Annals of Statistics

### 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

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