Linear programming problems for l_1- optimal frontier estimation

TL;DR

This paper introduces new linear programming-based kernel estimators for Lipschitz frontiers that are regular, minimal in support, and converge optimally in L_1 error to the true frontier.

Contribution

It presents a novel linear programming approach to construct Lipschitz frontier estimators with proven optimal convergence rates.

Findings

L_1 error converges almost surely to zero

Convergence rate is proven to be optimal

Estimators are regular and cover all points

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.