Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms

Thomas Mathew, Kenneth Nordström

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

For a normally distributed random matrixYwith a general variance-covariance matrixΣY, and for a nonnegative definite matrixQ, necessary and sufficient conditions are derived for the Wishartness ofY′QY. The conditions resemble those obtained by Wong, Masaro, and Wang (1991,J. Multivariate Anal.39, 154-174) and Wong and Wang (1993,J. Multivariate Anal.44, 146-159), but are verifiable and are obtained by elementary means. An explicit characterization is also obtained for the structure ofΣYunder which the distribution ofY′QYis Wishart. AssumingΣYpositive definite, a necessary and sufficient condition is derived for every univariate quadratic fromlY′QYlto be distributed as a multiple of a chi-square. For the caseQ=In, the corresponding structure ofΣYis identified. An explicit counterexample is constructed showing that Wishartness ofY′Yneed not follow when, for every vectorl, l′Y′Ylis distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969,Sankhyā31, 19-22). Application of the results to multivariate components of variance models is briefly indicated.

Original languageEnglish
Pages (from-to)129-143
Number of pages15
JournalJournal of Multivariate Analysis
Volume61
Issue number1
DOIs
Publication statusPublished - Apr 1997
Externally publishedYes

Fingerprint

Chi-square Distribution
Quadratic form
Chi-square
Counterexample
Wishart Distribution
Necessary Conditions
Components of Variance
Sufficient Conditions
Univariate
Non-negative
Model

Keywords

  • Complex covariance structure; group symmetry covariance model; multivariate components of variance model; skew-symmetric matrix

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Numerical Analysis
  • Statistics and Probability

Cite this

Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms. / Mathew, Thomas; Nordström, Kenneth.

In: Journal of Multivariate Analysis, Vol. 61, No. 1, 04.1997, p. 129-143.

Research output: Contribution to journalArticle

Mathew, Thomas ; Nordström, Kenneth. / Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms. In: Journal of Multivariate Analysis. 1997 ; Vol. 61, No. 1. pp. 129-143.
@article{c6557408757740d0886871abd8f4f725,
title = "Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms",
abstract = "For a normally distributed random matrixYwith a general variance-covariance matrixΣY, and for a nonnegative definite matrixQ, necessary and sufficient conditions are derived for the Wishartness ofY′QY. The conditions resemble those obtained by Wong, Masaro, and Wang (1991,J. Multivariate Anal.39, 154-174) and Wong and Wang (1993,J. Multivariate Anal.44, 146-159), but are verifiable and are obtained by elementary means. An explicit characterization is also obtained for the structure ofΣYunder which the distribution ofY′QYis Wishart. AssumingΣYpositive definite, a necessary and sufficient condition is derived for every univariate quadratic fromlY′QYlto be distributed as a multiple of a chi-square. For the caseQ=In, the corresponding structure ofΣYis identified. An explicit counterexample is constructed showing that Wishartness ofY′Yneed not follow when, for every vectorl, l′Y′Ylis distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969,Sankhyā31, 19-22). Application of the results to multivariate components of variance models is briefly indicated.",
keywords = "Complex covariance structure; group symmetry covariance model; multivariate components of variance model; skew-symmetric matrix",
author = "Thomas Mathew and Kenneth Nordstr{\"o}m",
year = "1997",
month = "4",
doi = "10.1006/jmva.1997.1665",
language = "English",
volume = "61",
pages = "129--143",
journal = "Journal of Multivariate Analysis",
issn = "0047-259X",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms

AU - Mathew, Thomas

AU - Nordström, Kenneth

PY - 1997/4

Y1 - 1997/4

N2 - For a normally distributed random matrixYwith a general variance-covariance matrixΣY, and for a nonnegative definite matrixQ, necessary and sufficient conditions are derived for the Wishartness ofY′QY. The conditions resemble those obtained by Wong, Masaro, and Wang (1991,J. Multivariate Anal.39, 154-174) and Wong and Wang (1993,J. Multivariate Anal.44, 146-159), but are verifiable and are obtained by elementary means. An explicit characterization is also obtained for the structure ofΣYunder which the distribution ofY′QYis Wishart. AssumingΣYpositive definite, a necessary and sufficient condition is derived for every univariate quadratic fromlY′QYlto be distributed as a multiple of a chi-square. For the caseQ=In, the corresponding structure ofΣYis identified. An explicit counterexample is constructed showing that Wishartness ofY′Yneed not follow when, for every vectorl, l′Y′Ylis distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969,Sankhyā31, 19-22). Application of the results to multivariate components of variance models is briefly indicated.

AB - For a normally distributed random matrixYwith a general variance-covariance matrixΣY, and for a nonnegative definite matrixQ, necessary and sufficient conditions are derived for the Wishartness ofY′QY. The conditions resemble those obtained by Wong, Masaro, and Wang (1991,J. Multivariate Anal.39, 154-174) and Wong and Wang (1993,J. Multivariate Anal.44, 146-159), but are verifiable and are obtained by elementary means. An explicit characterization is also obtained for the structure ofΣYunder which the distribution ofY′QYis Wishart. AssumingΣYpositive definite, a necessary and sufficient condition is derived for every univariate quadratic fromlY′QYlto be distributed as a multiple of a chi-square. For the caseQ=In, the corresponding structure ofΣYis identified. An explicit counterexample is constructed showing that Wishartness ofY′Yneed not follow when, for every vectorl, l′Y′Ylis distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969,Sankhyā31, 19-22). Application of the results to multivariate components of variance models is briefly indicated.

KW - Complex covariance structure; group symmetry covariance model; multivariate components of variance model; skew-symmetric matrix

UR - http://www.scopus.com/inward/record.url?scp=0037868442&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037868442&partnerID=8YFLogxK

U2 - 10.1006/jmva.1997.1665

DO - 10.1006/jmva.1997.1665

M3 - Article

VL - 61

SP - 129

EP - 143

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 1

ER -