L1L1-consistent estimation of the density of residuals in random design regression models

In this paper we study the problem of estimating the density of the error distribution in a random design regression model, where the error is assumed to be independent of the design variable. Our main result is that the L1L1 error of the kernel density estimate applied to residuals of a consistent regression estimate converges with probability 1 to 0 regardless of the form of the true density. We demonstrate that this result is in general no longer true if the error distribution and the design variable are dependent.