Summary Let Cb, b = 0, 1, …, B, be the class of all d-dimensional matrices (i.e. having d rows and d columns) with a block-diagonal structure, the blocks along the diagonal being the identity matrix of order d1b, and mjb repetitions of a symmetric dsb-dimensional positive-definite matrix σsb, s = 2, 3, …, Sb. One or more of the dsb's may be zero. The model specifies initially that observations Wb, b ⩾ 1, are independent Wishart matrices of nb degrees of freedom and expected value nb σb, σb ∈ Cb. The hypothesis H0 to be tested is that σb, b ⩾ 1, are all equal to a matrix σ0 in C0. It is assumed that C0⊂C1∩C2∩…∩CB. The likelihood-ratio test criterion is obtained. Several current multivariate test criteria are shown to be special cases, but they do not exhaust the cases that are useful. From the Mellin transform of the test criterion, a χ2 series, a χ2 approximation, a non-central χ2 series and a non-central χ2 approximation are derived for the null distribtion of the test criterion. The error in the non-central χ2 approximation is of the order of n−3b, while the error in the χ2 approximation is of the order of n−2b.
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