A Simple Consistent Non-parametric Estimator of the Regression Function in a Truncated Sample

Much recent work has focused on the estimation of regression functions in samples which are truncated or censored. Much of this work has focused on the estimation of a parametric regression function with an error distribution of unknown form. While these method relax a strong parametric assumption about which we seldom have a priori information, they still impose a strong parametric assumption on the regression equation (which is presumably the focus of the analysis). Here we take the other approach. An estimator is proposed for the problem of non-parametric regression when the sample is truncated above or below some known threshold of the dependent variable. We specify the error distribution up to a vector of parameters while estimating the regression function without assuming a parametric form. A simple ``backfit'' estimator based on an initial kernel smooth is proposed. We establish consistency results for this estimator when the error distribution is known up to a finite parameter vector and satisfies some regularity conditions. A small monte-carlo study is performed to ascertain the finite sample properties of the estimator. The estimator is found to perform well in our experiment: achieving reasonableaverage absolute errors relative to the maximum likelihood estimator- especially when truncation is severe.