Extreme value and Haar series estimates of point process boundaries

TL;DR

This paper introduces a novel method combining Haar series and extreme value theory to estimate the boundary of a 2D set from interior points, with convergence analysis and bias reduction techniques.

Contribution

It develops a new boundary estimation approach using Haar series and extreme values, providing convergence conditions and bias correction methods.

Findings

Different limit distributions identified for the estimator

Method effectively reduces negative bias in boundary estimates

Simulation demonstrates practical performance of the approach

Abstract

We present a new method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on Haar series and extreme values of the point process. We give conditions for various kind of convergence and we obtain remarkably different possible limit distributions. We propose a method of reducing the negative bias, illustrated by a simulation.