Statistica Sinica 5(1995), 373-397

ON SOME CENTRAL AND NON-CENTRAL MULTIVARIATE

CHI-SQUARE DISTRIBUTIONS

Thomas Royen

Fachhochschule Bingen

Abstract: Let R be a non-singular m-factorial correlation matrix, i.e. R=D+AA' with a diagonal matrix D>0 and a not necessarily definite matrix AA' of the minimal possible rank m. From an expression for the general non-central multivariate 2-distribution function with the accompan ying correlation matrix R some simpler cases are derived: The p-variate central 2-distribution with q degrees of freedom is given as a mixture with regard to a Wishart Wm(q, Im)-distribu tion. For m=2 several integral and series representations are derived including the limit case with exactly one zero on the diagonal of D. The two-factorial representation is applied to the four-variate 2-distribution. Besides, it is used for T aylor approximations if m>2. Furthermore, the non-central distribution function is given for m=1 and applied to power calculations for some multivariate multiple comparisons with a control.

Key words and phrases: Multivariate chi-square distribution, multivariate gamma distribution, multivariate Rayleigh distribution, multivariate multiple comparisons with a control, power.