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Large Sample Approximation of the Distribution for Convex‐Hull Estimators of Boundaries
Author(s) -
JEONG S.O.,
PARK B. U.
Publication year - 2006
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2006.00452.x
Subject(s) - mathematics , estimator , convex hull , boundary (topology) , convex conjugate , function (biology) , mathematical analysis , statistics , regular polygon , geometry , convex body , evolutionary biology , biology
. Given n independent and identically distributed observations in a set G = {( x , y ) ∈ [0, 1] p × 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 bias of the convex‐hull and the DEA estimators and to construct confidence intervals for the true function.