Statistica Sinica

Thomas Royen

Abstract:LetRbe a non-singularm-factorial correlation matrix, i.e.R=D+AA'with a diagonal matrixD>0 and a not necessarily definite matrixAA'of the minimal possible rankm. From an expression for the general non-central multivariateX-distribution function with the accompan ying correlation matrix^{2}Rsome simpler cases are derived: Thep-variate centralX-distribution with^{2}qdegrees of freedom is given as a mixture with regard to a WishartW(q,_{m}I_{m})-distribu tion. Form=2 several integral and series representations are derived including the limit case with exactly one zero on the diagonal ofD. The two-factorial representation is applied to the four-variateX-distribution. Besides, it is used for T aylor approximations if^{2}m>2. Furthermore, the non-central distribution function is given form=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.