A Missing Information Principle and $M$-Estimators in Regression Analysis with Censored and Truncated Data

Abstract

A general missing information principle is proposed for constructing $M$-estimators of regression parameters in the presence of left truncation and right censoring on the observed responses. By making use of martingale central limit theorems and empirical process theory, the asymptotic normality of $M$-estimators is established under certain assumptions. Asymptotically efficient $M$-estimators are also developed by using data-dependent score functions. In addition, robustness properties of the estimators are discussed and formulas for their influence functions are derived for the robustness analysis.