Summary: The \(p\times p\) real matrix \(\Sigma\) is called block-spherical if it is diagonal with \(q\) blocks, \(1\leq q\leq p\), each containing \(p_ i\) elements equal to \(\sigma^ 2_ i\), \(i=1,\dots,q\), \(p_ 1+p_ 2+\cdots+ p_ q=p\). The hypothesis of multisample block-sphericity is that \(k\) covariance matrices \(\Sigma_ j\) of \(p\)-variate normal populations are block-spherical and equal.
This paper considers the modified likelihood ratio test for multisample block-sphericity, its moments, and representations for its null and non- null distribution. In particular, it gives a series expansion for the non-null distribution under local alternatives. The paper extends previous results of the author [Am. J. Math. Manage. Sci. 8, No. 1/2, 135-163 (1988; Zbl 0726.62020)].