Asymptotic Properties of Error Density Estimators in the Two-Phase Linear Regression Model

This paper investigates kernel estimation of the error density function for the two-phase linear regression model. We derive the asymptotic distributions of residual-based kernel density estimators. First, we demonstrate that the asymptotic distribution of the maximum deviation (suitably normalized) between the residual-based kernel density estimator and the expected kernel density (based on the true errors) coincides with the result for an independent and identically distributed (i.i.d.) sample, as established in Bickel and Rosenblatt (1973). We then prove that the residual-based kernel density estimator is asymptotically normal at a fixed point.