## The Annals of Statistics

### Regression M-estimators with doubly censored data

#### Abstract

The M-estimators are proposed for the linear regression model with random design when the response observations are doubly censored. The proposed estimators are constructed as some functional of a Campbell-type estimator $\hat{F}_n$ for a bivariate distribution function based on data which are doubly censored in one coordinate. We establish strong uniform consistency and asymptotic normality of $\hat{F}_n$ and derive the asymptotic normality of the proposed regression M-estimators through verifying their Hadamard differentiability property. As corollaries, we show that our results on the proposed M-estimators also apply to other types of data such as uncensored observations, bivariate observations under univariate right censoring, bivariate right-censored observations, and so on. Computation of the proposed regression M-estimators is discussed and the method is applied to a doubly censored data set, which was encountered in a recent study on the age-dependent growth rate of primary breast cancer.

#### Article information

Source
Ann. Statist. Volume 25, Number 6 (1997), 2638-2664.

Dates
First available in Project Euclid: 30 August 2002

http://projecteuclid.org/euclid.aos/1030741089

Digital Object Identifier
doi:10.1214/aos/1030741089

Mathematical Reviews number (MathSciNet)
MR1604432

Zentralblatt MATH identifier
0907.62045

Subjects
Primary: 62G05: Estimation 62J05: Linear regression
Secondary: 62E20: Asymptotic distribution theory

#### Citation

Ren, Jian-Jian; Gu, Minggao. Regression M -estimators with doubly censored data. Ann. Statist. 25 (1997), no. 6, 2638--2664. doi:10.1214/aos/1030741089. http://projecteuclid.org/euclid.aos/1030741089.

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