Equivalence of smoothing parameter selectors in density and intensity estimation

Peter Diggle, J. S. Marron

Research output: Contribution to journalArticle

Abstract

Kernel smoothing is an attractive method for the nonparametric estimation of either a probability density function or the intensity function of a nonstationary Poisson process. In each case the amount of smoothing, controlled by the bandwidth, that is, smoothing parameter, is crucial to the performance of the estimator. Bandwidth selection by cross-validation has been widely studied in the context of density estimation. A bandwidth selector in the intensity estimation case has been proposed that minimizes an estimate of the mean squared error under the assumption that the data are generated by a stationary Cox process. This article shows that these two methods each select the same bandwidth, even though they are motivated in much different ways. In addition to providing further justification of each method, this equivalence of smoothing parameter selectors yields new insights for both density and intensity estimation. A benefit for intensity estimation is that this equivalence makes it clear how the Cox process method may be applied to kernels that are nonuniform, or even of higher order. Another benefit is that this duality between problems makes it clear how to apply the well-developed asymptotic methods for understanding density estimation in the intensity setting. A benefit for density estimation is that it motivates an analog of the Cox process method, which provides a useful nonasymptotic means of studying that problem. The specific forms of the estimators and smoothing parameter selectors are introduced in Section 1. The basic equivalence result is stated in Section 2. Sections 3 and 4 describe new insights that follow for intensity and density estimation, respectively. Section 5 discusses modification of these ideas to take boundary effects into consideration and shows how they can be used to motivate new boundary adjustments in intensity estimation.

Original languageEnglish (US)
Pages (from-to)793-800
Number of pages8
JournalJournal of the American Statistical Association
Volume83
Issue number403
DOIs
StatePublished - 1988
Externally publishedYes

Fingerprint

Selector
Smoothing Parameter
Equivalence
Density Estimation
Cox Process
Bandwidth
Estimator
Kernel Smoothing
Bandwidth Selection
Boundary Effect
Nonstationary Processes
Intensity Function
Asymptotic Methods
Nonparametric Estimation
Stationary Process
Poisson process
Mean Squared Error
Cross-validation
Justification
Probability density function

Keywords

  • Cox process
  • Cross-validation
  • Kernel estimators
  • Nonparametric density estimation
  • Poisson intensity estimation
  • Smoothing

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Equivalence of smoothing parameter selectors in density and intensity estimation. / Diggle, Peter; Marron, J. S.

In: Journal of the American Statistical Association, Vol. 83, No. 403, 1988, p. 793-800.

Research output: Contribution to journalArticle

@article{9919a68dfe8044c1946d7518fee961cc,
title = "Equivalence of smoothing parameter selectors in density and intensity estimation",
abstract = "Kernel smoothing is an attractive method for the nonparametric estimation of either a probability density function or the intensity function of a nonstationary Poisson process. In each case the amount of smoothing, controlled by the bandwidth, that is, smoothing parameter, is crucial to the performance of the estimator. Bandwidth selection by cross-validation has been widely studied in the context of density estimation. A bandwidth selector in the intensity estimation case has been proposed that minimizes an estimate of the mean squared error under the assumption that the data are generated by a stationary Cox process. This article shows that these two methods each select the same bandwidth, even though they are motivated in much different ways. In addition to providing further justification of each method, this equivalence of smoothing parameter selectors yields new insights for both density and intensity estimation. A benefit for intensity estimation is that this equivalence makes it clear how the Cox process method may be applied to kernels that are nonuniform, or even of higher order. Another benefit is that this duality between problems makes it clear how to apply the well-developed asymptotic methods for understanding density estimation in the intensity setting. A benefit for density estimation is that it motivates an analog of the Cox process method, which provides a useful nonasymptotic means of studying that problem. The specific forms of the estimators and smoothing parameter selectors are introduced in Section 1. The basic equivalence result is stated in Section 2. Sections 3 and 4 describe new insights that follow for intensity and density estimation, respectively. Section 5 discusses modification of these ideas to take boundary effects into consideration and shows how they can be used to motivate new boundary adjustments in intensity estimation.",
keywords = "Cox process, Cross-validation, Kernel estimators, Nonparametric density estimation, Poisson intensity estimation, Smoothing",
author = "Peter Diggle and Marron, {J. S.}",
year = "1988",
doi = "10.1080/01621459.1988.10478665",
language = "English (US)",
volume = "83",
pages = "793--800",
journal = "Journal of the American Statistical Association",
issn = "0162-1459",
publisher = "Taylor and Francis Ltd.",
number = "403",

}

TY - JOUR

T1 - Equivalence of smoothing parameter selectors in density and intensity estimation

AU - Diggle, Peter

AU - Marron, J. S.

PY - 1988

Y1 - 1988

N2 - Kernel smoothing is an attractive method for the nonparametric estimation of either a probability density function or the intensity function of a nonstationary Poisson process. In each case the amount of smoothing, controlled by the bandwidth, that is, smoothing parameter, is crucial to the performance of the estimator. Bandwidth selection by cross-validation has been widely studied in the context of density estimation. A bandwidth selector in the intensity estimation case has been proposed that minimizes an estimate of the mean squared error under the assumption that the data are generated by a stationary Cox process. This article shows that these two methods each select the same bandwidth, even though they are motivated in much different ways. In addition to providing further justification of each method, this equivalence of smoothing parameter selectors yields new insights for both density and intensity estimation. A benefit for intensity estimation is that this equivalence makes it clear how the Cox process method may be applied to kernels that are nonuniform, or even of higher order. Another benefit is that this duality between problems makes it clear how to apply the well-developed asymptotic methods for understanding density estimation in the intensity setting. A benefit for density estimation is that it motivates an analog of the Cox process method, which provides a useful nonasymptotic means of studying that problem. The specific forms of the estimators and smoothing parameter selectors are introduced in Section 1. The basic equivalence result is stated in Section 2. Sections 3 and 4 describe new insights that follow for intensity and density estimation, respectively. Section 5 discusses modification of these ideas to take boundary effects into consideration and shows how they can be used to motivate new boundary adjustments in intensity estimation.

AB - Kernel smoothing is an attractive method for the nonparametric estimation of either a probability density function or the intensity function of a nonstationary Poisson process. In each case the amount of smoothing, controlled by the bandwidth, that is, smoothing parameter, is crucial to the performance of the estimator. Bandwidth selection by cross-validation has been widely studied in the context of density estimation. A bandwidth selector in the intensity estimation case has been proposed that minimizes an estimate of the mean squared error under the assumption that the data are generated by a stationary Cox process. This article shows that these two methods each select the same bandwidth, even though they are motivated in much different ways. In addition to providing further justification of each method, this equivalence of smoothing parameter selectors yields new insights for both density and intensity estimation. A benefit for intensity estimation is that this equivalence makes it clear how the Cox process method may be applied to kernels that are nonuniform, or even of higher order. Another benefit is that this duality between problems makes it clear how to apply the well-developed asymptotic methods for understanding density estimation in the intensity setting. A benefit for density estimation is that it motivates an analog of the Cox process method, which provides a useful nonasymptotic means of studying that problem. The specific forms of the estimators and smoothing parameter selectors are introduced in Section 1. The basic equivalence result is stated in Section 2. Sections 3 and 4 describe new insights that follow for intensity and density estimation, respectively. Section 5 discusses modification of these ideas to take boundary effects into consideration and shows how they can be used to motivate new boundary adjustments in intensity estimation.

KW - Cox process

KW - Cross-validation

KW - Kernel estimators

KW - Nonparametric density estimation

KW - Poisson intensity estimation

KW - Smoothing

UR - http://www.scopus.com/inward/record.url?scp=0000237857&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000237857&partnerID=8YFLogxK

U2 - 10.1080/01621459.1988.10478665

DO - 10.1080/01621459.1988.10478665

M3 - Article

AN - SCOPUS:0000237857

VL - 83

SP - 793

EP - 800

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

IS - 403

ER -