Large sample approximation of the distribution for convex-hull estimators of boundaries

Given n independent and identically distributed observations in a set G = {(x, y) is an element of vertical bar 0, 1 vertical bar(p) x R : 0 <= y <= g(x)} with an unknown function g, called a boundary or frontier, it is desired to estimate g from the observations. The problem has several important applications including classification and cluster analysis, and is closely related to edge estimation in image reconstruction. The convex-hull estimator of a boundary or frontier is also very popular in econometrics, where it is a cornerstone of a method known as 'data envelope analysis'. In this paper, we give a large sample approximation of the distribution of the convex-hull estimator in the general case where p >= 1. We discuss ways of using the large sample approximation to correct the bins of the convex-hull and the DEA estimators and to construct confidence intervals for the true function.