Large Sample Theory of Estimation in Biased Sampling Regression Models. I

Abstract

Biased sampling regression models were introduced by Jewell, generalizing the truncated regression model studied by Bhattacharya, Chernoff and Yang. If the independent variable takes on only a finite number of values (as does the stratum variable), we show: 1. That if the slope of the underlying regression model is assumed known, then the nonparametric maximum likelihood estimates of the distribution of the independent and dependent variables (a) can be calculated from ordinary $M$ estimates; (b) are asymptotically efficient. 2. How to construct $M$ estimates of the slope which are always $\sqrt n$ consistent, asymptotically Gaussian and are efficient locally, for example, if the error distribution is Gaussian. We support our asymptotics with a small simulation.