Multivariate gamma distributions with one-factorial accompanying correlation matrices and applications to the distribution of the multivariate range

Abstract

A new representation for the characteristic function of the joint distribution of the Mahalanobis distances betweenk independentN(μ, Σ)-distributed points is given. Especially fork=3 the corresponding distribution function is obtained as a special case of multivariate gamma distributions whose accompanying normal distribution has a positive semidefinite correlation matrix with correlationsϱ ij=−a i a j. These gamma distribution functions are given here by one-dimensional parameter integrals. With some further trivariate gamma distributions third order Bonferroni inequalities are derived for the upper tails of the distribution function of the multivariate range ofk independentN(μ, I)-distributed points. From these inequalities very accurate (conservative) approximations to upperα-level bounds can also be computed for studentized multivariate ranges.