A note on extreme values and kernel estimators of sample boundaries

TL;DR

This paper discusses kernel estimators for the boundaries of sample sets, focusing on extreme values and extending previous work from Poisson point processes to random samples, providing new insights and methods.

Contribution

It offers essential insights and a method to extend kernel boundary estimation from Poisson processes to random samples, enhancing understanding of extreme value-based estimators.

Findings

Provides a method to relate Poisson process estimates to random sample estimates.

Extends previous boundary estimation techniques to more general sampling scenarios.

Offers theoretical insights into the behavior of kernel estimators at sample boundaries.

Abstract

In a previous paper, we studied a kernel estimate of the upper edge of a two-dimensional bounded set, based upon the extreme values of a Poisson point process. The initial paper "Geffroy J. (1964) Sur un probl\`eme d'estimation g\'eom\'etrique.Publications de l'Institut de Statistique de l'Universit\'e de Paris, XIII, 191-200" on the subject treats the frontier as the boundary of the support set for a density and the points as a random sample. We claimed in"Girard, S. and Jacob, P. (2004) Extreme values and kernel estimates of point processes boundaries.ESAIM: Probability and Statistics, 8, 150-168" that we are able to deduce the random sample case fr om the point process case. The present note gives some essential indications to this end, including a method which can be of general interest.