A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis

Ronald L. Horswell, Stephen W. Looney

Research output: Contribution to journalArticle

34 Citations (Scopus)

Abstract

The examination of coefficients of multivariate skewness and kurtosis is one of the more commonly used techniques for assessing multivariate normality (MVN). In this article, several tests for MVN based on these coefficients are compared via Monte Carlo simulation. The tests considered here include those based on Mardia’s affine-invariant measures of multivariate skewness and kurtosis and an omnibus procedure that combines the two. Also included are Small's tests, which are based on combinations of the marginal skewness and kurtosis coefficients and are coordinate-dependent. These tests are compared in terms of their power against a wide variety of non-MVN distributions; included in these alternatives are distributions with iid components, as well as distributions with positively-correlated components. Among the alternatives considered are non-MVN distributions with skewed components, symmetric components, univariate normal components, and MVN values of skewness and kurtosis. The tests considered here perform as expected; the affine-invariant procedures outperform the coordinate-de-pendent ones when the variables are correlated, whereas the coordinate-dependent tests are more powerful when the variables are uncorrelated. Neither type of procedure performs well when the alternative distribution has MVN values of skewness and kurtosis. The practical implications of these results for assessing MVN, as well as a method for combining the two types of tests, are discussed. The limitations of the use of coefficients of multivariate skewness and kurtosis in “diagnosing” departures from MVN are also described.

Original languageEnglish (US)
Pages (from-to)21-38
Number of pages18
JournalJournal of Statistical Computation and Simulation
Volume42
Issue number1-2
DOIs
StatePublished - Aug 1 1992

Fingerprint

Multivariate Normality
Kurtosis
Skewness
Affine Invariant
Coefficient
Normality
Alternatives
Dependent
Monte Carlo simulation
Invariant Measure
Univariate
Monte Carlo Simulation

Keywords

  • Affine-invariant
  • Khintchine distribution
  • Monte Carlo simulation
  • coordinate-dependent
  • generalized exponential power distribution
  • power studies

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

Cite this

@article{56cc68bc554d4c4a8025409ab8ff4e0b,
title = "A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis",
abstract = "The examination of coefficients of multivariate skewness and kurtosis is one of the more commonly used techniques for assessing multivariate normality (MVN). In this article, several tests for MVN based on these coefficients are compared via Monte Carlo simulation. The tests considered here include those based on Mardia’s affine-invariant measures of multivariate skewness and kurtosis and an omnibus procedure that combines the two. Also included are Small's tests, which are based on combinations of the marginal skewness and kurtosis coefficients and are coordinate-dependent. These tests are compared in terms of their power against a wide variety of non-MVN distributions; included in these alternatives are distributions with iid components, as well as distributions with positively-correlated components. Among the alternatives considered are non-MVN distributions with skewed components, symmetric components, univariate normal components, and MVN values of skewness and kurtosis. The tests considered here perform as expected; the affine-invariant procedures outperform the coordinate-de-pendent ones when the variables are correlated, whereas the coordinate-dependent tests are more powerful when the variables are uncorrelated. Neither type of procedure performs well when the alternative distribution has MVN values of skewness and kurtosis. The practical implications of these results for assessing MVN, as well as a method for combining the two types of tests, are discussed. The limitations of the use of coefficients of multivariate skewness and kurtosis in “diagnosing” departures from MVN are also described.",
keywords = "Affine-invariant, Khintchine distribution, Monte Carlo simulation, coordinate-dependent, generalized exponential power distribution, power studies",
author = "Horswell, {Ronald L.} and Looney, {Stephen W.}",
year = "1992",
month = "8",
day = "1",
doi = "10.1080/00949659208811407",
language = "English (US)",
volume = "42",
pages = "21--38",
journal = "Journal of Statistical Computation and Simulation",
issn = "0094-9655",
publisher = "Taylor and Francis Ltd.",
number = "1-2",

}

TY - JOUR

T1 - A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis

AU - Horswell, Ronald L.

AU - Looney, Stephen W.

PY - 1992/8/1

Y1 - 1992/8/1

N2 - The examination of coefficients of multivariate skewness and kurtosis is one of the more commonly used techniques for assessing multivariate normality (MVN). In this article, several tests for MVN based on these coefficients are compared via Monte Carlo simulation. The tests considered here include those based on Mardia’s affine-invariant measures of multivariate skewness and kurtosis and an omnibus procedure that combines the two. Also included are Small's tests, which are based on combinations of the marginal skewness and kurtosis coefficients and are coordinate-dependent. These tests are compared in terms of their power against a wide variety of non-MVN distributions; included in these alternatives are distributions with iid components, as well as distributions with positively-correlated components. Among the alternatives considered are non-MVN distributions with skewed components, symmetric components, univariate normal components, and MVN values of skewness and kurtosis. The tests considered here perform as expected; the affine-invariant procedures outperform the coordinate-de-pendent ones when the variables are correlated, whereas the coordinate-dependent tests are more powerful when the variables are uncorrelated. Neither type of procedure performs well when the alternative distribution has MVN values of skewness and kurtosis. The practical implications of these results for assessing MVN, as well as a method for combining the two types of tests, are discussed. The limitations of the use of coefficients of multivariate skewness and kurtosis in “diagnosing” departures from MVN are also described.

AB - The examination of coefficients of multivariate skewness and kurtosis is one of the more commonly used techniques for assessing multivariate normality (MVN). In this article, several tests for MVN based on these coefficients are compared via Monte Carlo simulation. The tests considered here include those based on Mardia’s affine-invariant measures of multivariate skewness and kurtosis and an omnibus procedure that combines the two. Also included are Small's tests, which are based on combinations of the marginal skewness and kurtosis coefficients and are coordinate-dependent. These tests are compared in terms of their power against a wide variety of non-MVN distributions; included in these alternatives are distributions with iid components, as well as distributions with positively-correlated components. Among the alternatives considered are non-MVN distributions with skewed components, symmetric components, univariate normal components, and MVN values of skewness and kurtosis. The tests considered here perform as expected; the affine-invariant procedures outperform the coordinate-de-pendent ones when the variables are correlated, whereas the coordinate-dependent tests are more powerful when the variables are uncorrelated. Neither type of procedure performs well when the alternative distribution has MVN values of skewness and kurtosis. The practical implications of these results for assessing MVN, as well as a method for combining the two types of tests, are discussed. The limitations of the use of coefficients of multivariate skewness and kurtosis in “diagnosing” departures from MVN are also described.

KW - Affine-invariant

KW - Khintchine distribution

KW - Monte Carlo simulation

KW - coordinate-dependent

KW - generalized exponential power distribution

KW - power studies

UR - http://www.scopus.com/inward/record.url?scp=0002137108&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002137108&partnerID=8YFLogxK

U2 - 10.1080/00949659208811407

DO - 10.1080/00949659208811407

M3 - Article

AN - SCOPUS:0002137108

VL - 42

SP - 21

EP - 38

JO - Journal of Statistical Computation and Simulation

JF - Journal of Statistical Computation and Simulation

SN - 0094-9655

IS - 1-2

ER -