Non-null distributions of the likelihood ratio criteria for independence and equality of mean vectors and covariance matrices

1972 ◽  
Vol 24 (1) ◽  
pp. 67-79 ◽  
Author(s):  
Hisao Nagao
Keyword(s):  
Likelihood Ratio ◽  
Covariance Matrices ◽  
Null Distributions ◽  
Mean Vectors ◽  
Equality Of Mean Vectors

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

2016 ◽  
Vol 2016 ◽  
pp. 1-25 ◽  
Author(s):  
Carlos A. Coelho ◽  
Filipe J. Marques ◽  
Sandra Oliveira
Keyword(s):  
Covariance Matrix ◽  
Likelihood Ratio ◽  
Exact Distribution ◽  
Covariance Matrices ◽  
Distribution Functions ◽  
Small Samples ◽  
Chi Square ◽  
Exact Distributions ◽  
Mean Vectors ◽  
Likelihood Ratio Statistics

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.

Type I Error Rates for Yao’s and James Tests of Equality of Mean Vectors Under VarianceCovariance Heteroscedasticity

1988 ◽  
Vol 13 (3) ◽  
pp. 281-290 ◽  
Author(s):  
James Algina ◽  
Kezhen L. Tang
Keyword(s):  
Sample Size ◽  
Type I Error ◽  
Error Rates ◽  
Covariance Matrices ◽  
Type I ◽  
Type I Error Rates ◽  
Mean Vectors ◽  
Equality Of Mean Vectors

For Yao’s and James’ tests, Type I error rates were estimated for various combinations of the number of variables (p), samplesize ratio (n1: n2), sample-size-to-variables ratio, and degree of heteroscedasticity. These tests are alternatives to Hotelling’s T2 and are intended for use when the variance-covariance matrices are not equal in a study using two independent samples. The performance of Yao’s test was superior to that of James’. Yao’s test had appropriate Type I error rates when p ≥ 10, (n1 + n2)/p ≥ 10, and 1:2 ≤ n1:n2 ≤ 2:1. When (n1 + n2)/p = 20, Yao’s test was robust when n1: n2 was 5:1, 3:1, and 4:1 and p was 2, 6, and 10, respectively.

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