On Some Central and Non - Central Multivariate Chi - Square Distributions

Let R be a non-singular m-factorial correlation matrix, i.e. R = D + AA0 with a diagonal matrixD > 0 and a not necessarily de nite matrix AA0 of the minimal possible rank m. From an expression for the general non-central multivariate distribution function with the accompanying correlation matrix R some simpler cases are derived: The p-variate central -distribution with q degrees of freedom is given as a mixture with regard to a Wishart Wm(q; Im)-distribution. 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 -distribution. Besides, it is used for Taylor 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.