Linear programming problems for l_1- optimal frontier estimation

Abstract

We propose new optimal estimators for the Lipschitz frontier of a set of points. They are defined as kernel estimators being sufficiently regular, covering all the points and whose associated support is of smallest surface. The estimators are written as linear combinations of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving related linear programming problem. The L_1 error between the estimated and the true frontier function with a known Lipschitz constant is shown to be almost surely converging to zero, and the rate of convergence is proved to be optimal.