Abstract
The multivariate Studentized range Rmax is defined as the positive square root of the maximum of generalized Studentized distances between any two points in a random sample drawn from a multivariate distribution. A short note on the background of Rmax is given. Then, for obtaining good approximation to the exact upper percentiles of Rmax the modified second approximation method is explained. Under the assumption of normality, the asymptotic expansion formulas necesary to evaluate those percentiles are drived for the large number of drgrees of freedom of the sample covariance matrix. The accuracy of the estimated percentiles is investigated extensively based on Monte Carlo studies. In the studies the simulated distributions of Rmax are calculated for selected values of parameters, some of which are presented in this paper. The results obtained by numerical examinations are summarized, in which the valid ranges of parameters for the formulas of the modified second approximation are discussed. Finally, simultaneous pairwise comparesions of mean vectors of m mulitinormal populations are discussed as an example of the use of Rmax.
