Estimation of regression parameters with left truncated data

Estimation in the linear regression model Y = beta'Z + epsilon is considered for the left truncated data, in which observations of the dependent variable Y are interfered by another random variable T in such a way that both Y and T are observable only if Y greater than or equal to T. We construct certain weighted least-square estimates (β) over cap for the regression parameter beta where the weights are random quantities determined by the product-limit estimates of the distribution function of T. Estimators for the error variance and the error distribution F-epsilon of epsilon are also provided. In the construction and the proofs of strong consistency and asymptotic normality of the estimators, a special estimator of the truncation probability P[Y greater than or equal to T] plays a major role. Our results are obtained under very weak conditions in which the joint distributions of Y and T are left arbitrary. Our estimation procedure is non-iterative which is much less computationally demanding than the iterative procedures available in the literature. Counterexamples are provided to show that a certain boundary condition on the underlying distributions is necessary for ensuring the identifiability of the regression parameters. (C) 2002 Elsevier B.V. All rights reserved.