Distribution of the Λ-Criterion for Sphericity Test and Its Connection to a Generalized Dirichlet Model

Abstract

It is shown that the exact null distribution of the likelihood ratio criterion for sphericity test in the p-variate normal case and the marginal distribution of the first component of a (p − 1)-variate generalized Dirichlet model with a given set of parameters are identical. The exact distribution of the likelihood ratio criterion so obtained has a general format for every p. A novel idea is introduced here through which the complicated exact null distribution of the sphericity test criterion in multivariate statistical analysis is converted into an easily tractable marginal density in a generalized Dirichlet model. It provides a direct and easiest method of computation of p-values. The computation of p-values and a table of critical points corresponding to p = 3 and 4 are also presented.

Keywords:

• Beta variables
• Exact distribution
• Generalized Dirichlet density
• Likelihood ratio criterion
• Meijer's G-function
• Sphericity test

Mathematics Subject Classification:

• 62H10
• 62H15
• 60E05

Notes

p-values are simulated independently using SAS

* deviates considerably from our value and that may be due to typographical error.