TY - JOUR

T1 - Near-Exact Distributions for Likelihood Ratio Statistics Used in the Simultaneous Test of Conditions on Mean Vectors and Patterns of Covariance Matrices

AU - Coelho, Carlos A.

AU - Marques, Filipe J.

AU - Oliveira, Sandra

N1 - Fundacao para a Ciencia e a Tecnologia (FCT) - UID/MAT/00297/2013

PY - 2016

Y1 - 2016

N2 - The authors address likelihood ratio statistics used to test simultaneously conditions on mean vectors and patterns on covariance matrices. Tests for conditions on mean vectors, assuming or not a given structure for the covariance matrix, are quite common, since they may be easily implemented. But, on the other hand, the practical use of simultaneous tests for conditions on the mean vectors and a given pattern for the covariance matrix is usually hindered by the nonmanageability of the expressions for their exact distribution functions. The authors show the importance of being able to adequately factorize the c.f. of the logarithm of likelihood ratio statistics in order to obtain sharp and highly manageable near-exact distributions, or even the exact distribution in a highly manageable form. The tests considered are the simultaneous tests of equality or nullity of means and circularity, compound symmetry, or sphericity of the covariance matrix. Numerical studies show the high accuracy of the near-exact distributions and their adequacy for cases with very small samples and/or large number of variables. The exact and near-exact quantiles computed show how the common chi-square asymptotic approximation is highly inadequate for situations with small samples or large number of variables.

AB - The authors address likelihood ratio statistics used to test simultaneously conditions on mean vectors and patterns on covariance matrices. Tests for conditions on mean vectors, assuming or not a given structure for the covariance matrix, are quite common, since they may be easily implemented. But, on the other hand, the practical use of simultaneous tests for conditions on the mean vectors and a given pattern for the covariance matrix is usually hindered by the nonmanageability of the expressions for their exact distribution functions. The authors show the importance of being able to adequately factorize the c.f. of the logarithm of likelihood ratio statistics in order to obtain sharp and highly manageable near-exact distributions, or even the exact distribution in a highly manageable form. The tests considered are the simultaneous tests of equality or nullity of means and circularity, compound symmetry, or sphericity of the covariance matrix. Numerical studies show the high accuracy of the near-exact distributions and their adequacy for cases with very small samples and/or large number of variables. The exact and near-exact quantiles computed show how the common chi-square asymptotic approximation is highly inadequate for situations with small samples or large number of variables.

KW - NTEGER GAMMA-DISTRIBUTION

KW - MULTIVARIATE-ANALYSIS

UR - http://www.scopus.com/inward/record.url?scp=84964744581&partnerID=8YFLogxK

U2 - 10.1155/2016/8975902

DO - 10.1155/2016/8975902

M3 - Article

AN - SCOPUS:84964744581

VL - 2016

JO - Mathematical Problems in Engineering

JF - Mathematical Problems in Engineering

SN - 1024-123X

M1 - 8975902

ER -