A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ2 reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ2 reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795-800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.

Original languageEnglish (US)
Pages (from-to)233-240
Number of pages8
JournalStatistics and Computing
Volume18
Issue number3
DOIs
StatePublished - Sep 2008
Externally publishedYes

Fingerprint

Normal Mixture
Likelihood Ratio Test
Mixture Distribution
Gaussian distribution
Alternatives
Sample Size
Bootstrap Test
Parametric Bootstrap
Likelihood Ratio Statistic
Variance Components
Regularity Conditions
Unequal
Null hypothesis
Null
Likelihood
Degree of freedom
Restriction
Testing
Likelihood ratio test
Statistics

Keywords

  • Bootstrap
  • EM algorithm
  • Likelihood ratio test
  • Normal mixture

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Statistics and Probability

Cite this

A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture. / Lo, Yungtai.

In: Statistics and Computing, Vol. 18, No. 3, 09.2008, p. 233-240.

Research output: Contribution to journalArticle

@article{7b7d2ae5db0840129d8dc4b7627eced1,
title = "A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture",
abstract = "It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ2 reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ2 reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795-800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.",
keywords = "Bootstrap, EM algorithm, Likelihood ratio test, Normal mixture",
author = "Yungtai Lo",
year = "2008",
month = "9",
doi = "10.1007/s11222-008-9052-4",
language = "English (US)",
volume = "18",
pages = "233--240",
journal = "Statistics and Computing",
issn = "0960-3174",
publisher = "Springer Netherlands",
number = "3",

}

TY - JOUR

T1 - A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

AU - Lo, Yungtai

PY - 2008/9

Y1 - 2008/9

N2 - It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ2 reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ2 reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795-800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.

AB - It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ2 reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ2 reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795-800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.

KW - Bootstrap

KW - EM algorithm

KW - Likelihood ratio test

KW - Normal mixture

UR - http://www.scopus.com/inward/record.url?scp=45449093770&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45449093770&partnerID=8YFLogxK

U2 - 10.1007/s11222-008-9052-4

DO - 10.1007/s11222-008-9052-4

M3 - Article

VL - 18

SP - 233

EP - 240

JO - Statistics and Computing

JF - Statistics and Computing

SN - 0960-3174

IS - 3

ER -