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June, 1992 Large Sample Study of Empirical Distributions in a Random-Multiplicative Censoring Model
Y. Vardi, Cun-Hui Zhang
Ann. Statist. 20(2): 1022-1039 (June, 1992). DOI: 10.1214/aos/1176348668

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.

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Y. Vardi. Cun-Hui Zhang. "Large Sample Study of Empirical Distributions in a Random-Multiplicative Censoring Model." Ann. Statist. 20 (2) 1022 - 1039, June, 1992. https://doi.org/10.1214/aos/1176348668

Information

Published: June, 1992
First available in Project Euclid: 12 April 2007

zbMATH: 0761.62056
MathSciNet: MR1165604
Digital Object Identifier: 10.1214/aos/1176348668

Subjects:
Primary: 62G05
Secondary: 62G20

Rights: Copyright © 1992 Institute of Mathematical Statistics

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Vol.20 • No. 2 • June, 1992
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