Asymptotic Distribution of the Likelihood Ratio Test Statistic for Sphericity of Complex Multivariate Normal Distribution.

1981 ◽  
Author(s):  
C. Fang ◽  
P. R. Krishnaiah ◽  
B. N. Nagarsenker
Keyword(s):  
Normal Distribution ◽  
Asymptotic Distribution ◽  
Likelihood Ratio ◽  
Likelihood Ratio Test ◽  
Multivariate Normal Distribution ◽  
Likelihood Ratio Test Statistic ◽  
Ratio Test ◽  
Multivariate Normal ◽  
Test Statistic

A Note on the Asymptotic Distribution of Likelihood Ratio Tests to Test Variance Components

2006 ◽  
Vol 9 (4) ◽  
pp. 490-495 ◽  
Author(s):  
Peter M. Visscher
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Distribution ◽  
Likelihood Ratio ◽  
Likelihood Ratio Test ◽  
Variance Components ◽  
Degrees Of Freedom ◽  
Likelihood Ratio Tests ◽  
Likelihood Ratio Test Statistic ◽  
Ratio Test ◽  
Test Statistic

AbstractWhen using maximum likelihood methods to estimate genetic and environmental components of (co)variance, it is common to test hypotheses using likelihood ratio tests, since such tests have desirable asymptotic properties. In particular, the standard likelihood ratio test statistic is assumed asymptotically to follow a χ2 distribution with degrees of freedom equal to the number of parameters tested. Using the relationship between least squares and maximum likelihood estimators for balanced designs, it is shown why the asymptotic distribution of the likelihood ratio test for variance components does not follow a χ2 distribution with degrees of freedom equal to the number of parameters tested when the null hypothesis is true. Instead, the distribution of the likelihood ratio test is a mixture of χ2 distributions with different degrees of freedom. Implications for testing variance components in twin designs and for quantitative trait loci mapping are discussed. The appropriate distribution of the likelihood ratio test statistic should be used in hypothesis testing and model selection.

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