On the estimation of a support curve of indeterminate sharpness

Peter Hall, Michael Nussbaum, Steven E. Stern

Research output: Contribution to journalArticleResearchpeer-review

28 Citations (Scopus)

Abstract

We propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focused on cases where the rate of decrease is relatively fast, this being the most difficult setting. It demands the use of a relatively large number of bivariate order statistics. By way of comparison, support curve estimation in the context of slow rates of decrease of the density may be addressed using methods that employ only a relatively small number of order statistics at the extremities of the point cloud. In this paper we suggest a new type of estimator, based on projecting onto an axis those data values lying within a thin rectangular strip. Adaptive univariate methods are then applied to the problem of estimating an endpoint of the distribution on the axis. The new method is shown to have theoretically optimal performance in a range of settings. Its numerical properties are explored in a simulation study.

Original languageEnglish
Pages (from-to)204-232
Number of pages29
JournalJournal of Multivariate Analysis
Volume62
Issue number2
DOIs
Publication statusPublished - Aug 1997
Externally publishedYes

Fingerprint

Sharpness
Statistics
Order Statistics
Decrease
Curve
Curve Estimation
Nonparametric Methods
Point Cloud
Univariate
Strip
Simulation Study
Vary
Estimator
Range of data
Order statistics

Cite this

Hall, Peter ; Nussbaum, Michael ; Stern, Steven E. / On the estimation of a support curve of indeterminate sharpness. In: Journal of Multivariate Analysis. 1997 ; Vol. 62, No. 2. pp. 204-232.
@article{5a3f704f21e34551af58d23c52a0ef82,
title = "On the estimation of a support curve of indeterminate sharpness",
abstract = "We propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focused on cases where the rate of decrease is relatively fast, this being the most difficult setting. It demands the use of a relatively large number of bivariate order statistics. By way of comparison, support curve estimation in the context of slow rates of decrease of the density may be addressed using methods that employ only a relatively small number of order statistics at the extremities of the point cloud. In this paper we suggest a new type of estimator, based on projecting onto an axis those data values lying within a thin rectangular strip. Adaptive univariate methods are then applied to the problem of estimating an endpoint of the distribution on the axis. The new method is shown to have theoretically optimal performance in a range of settings. Its numerical properties are explored in a simulation study.",
author = "Peter Hall and Michael Nussbaum and Stern, {Steven E.}",
year = "1997",
month = "8",
doi = "10.1006/jmva.1997.1681",
language = "English",
volume = "62",
pages = "204--232",
journal = "Journal of Multivariate Analysis",
issn = "0047-259X",
publisher = "Academic Press Inc.",
number = "2",

}

On the estimation of a support curve of indeterminate sharpness. / Hall, Peter; Nussbaum, Michael; Stern, Steven E.

In: Journal of Multivariate Analysis, Vol. 62, No. 2, 08.1997, p. 204-232.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - On the estimation of a support curve of indeterminate sharpness

AU - Hall, Peter

AU - Nussbaum, Michael

AU - Stern, Steven E.

PY - 1997/8

Y1 - 1997/8

N2 - We propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focused on cases where the rate of decrease is relatively fast, this being the most difficult setting. It demands the use of a relatively large number of bivariate order statistics. By way of comparison, support curve estimation in the context of slow rates of decrease of the density may be addressed using methods that employ only a relatively small number of order statistics at the extremities of the point cloud. In this paper we suggest a new type of estimator, based on projecting onto an axis those data values lying within a thin rectangular strip. Adaptive univariate methods are then applied to the problem of estimating an endpoint of the distribution on the axis. The new method is shown to have theoretically optimal performance in a range of settings. Its numerical properties are explored in a simulation study.

AB - We propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focused on cases where the rate of decrease is relatively fast, this being the most difficult setting. It demands the use of a relatively large number of bivariate order statistics. By way of comparison, support curve estimation in the context of slow rates of decrease of the density may be addressed using methods that employ only a relatively small number of order statistics at the extremities of the point cloud. In this paper we suggest a new type of estimator, based on projecting onto an axis those data values lying within a thin rectangular strip. Adaptive univariate methods are then applied to the problem of estimating an endpoint of the distribution on the axis. The new method is shown to have theoretically optimal performance in a range of settings. Its numerical properties are explored in a simulation study.

UR - http://www.scopus.com/inward/record.url?scp=0031208941&partnerID=8YFLogxK

U2 - 10.1006/jmva.1997.1681

DO - 10.1006/jmva.1997.1681

M3 - Article

VL - 62

SP - 204

EP - 232

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -