Efficient L1 estimation and related inferences in linear regression with unknown form of heteroscedasticity

In linear regression analysis, L-1 estimation of regression parameters is a viable alternative to the least squares estimation. It is an extension of the concept of median, thereby having certain desirable robustness properties, and can be easily implemented via linear programming (Koenker and Bassett, 1978). Like the least squares method, it can lose efficiency when error terms are heteroscedastic. Koenker and Zhao (1994) showed that, when a parametric form of heteroscedasticity is assumed, one can obtain asymptotically efficient L-1 estimator by reweighting the regression equation, where the optimal weights are consistently estimated. In this article, we consider efficient L-1 estimation when the form of heteroscedasticity is unknown. We propose a sample-splitting method to construct consistent estimates for the weights and to avoid bias. We make use of a recently developed resampling method to approximate sampling distribution of the resulting estimate. Simulation results which provide assessment of efficiency gain of the proposed estimate as well as accuracy of the inference procedure are reported.