Abstract: | We define the Wishart distribution on the cone of positive definite matrices and an exponential distribution on the Lorentz cone as exponential dispersion models. We show that these two distributions possess a property of exact decomposition, and we use this property to solve the following problem: given q samples (yil,… yiNj), i = l,…,q, from a N(μi,Σi,) distribution, test H:Σ1 = Σ2 = … = σq. Using the exact decomposition property, the classical test statistic for H, involving q parameters pi = (Ni, - l)/2, i = 1,…,q, is replaced by a sequence of q - l test statistics for the sequence of tests Hi,:σ1 =σ2 = … =σi given that Hi-1 is true, i = 2,…,q. Each one of these test statistics involves two parameters only, p.i-1 = p1 + … + pi-1 and pi. We also use the exact decomposition property to test equality of the “direction parameters” for q sample points from the exponential distribution on the Lorentz cone. We give a table of critical values for the distribution on the three-dimensional Lorentz cone. Tables of critical values in higher dimensions can easily be computed following the same method as in dimension three. |