Optimum invariant tests on discriminant coefficients or means of multinormal population with additional information (Q788432)

This paper deals with cases where a sample is available for a p-dimensional normal population \((p=p_ 1+p_ 2)\), and an additional independent sample for the \(p_ 2\)-dimensional subpopulation. Hence, \(X_ 1,...,X_ N\) are i.i.d. \(N_ p(\mu,\Sigma)\), while \(W_ 1,...,W_ M\) are i.i.d. \(N_{p_ 2}(\mu_ 2,\Sigma_{22})\), independent of the \(X_ j\). Here \(\mu^ t=(\mu_ 1,\mu_ 2)\), and \(\Sigma\) is partitioned to conform. The following hypotheses are tested: \(H_ 0:\Gamma_ 1=0\), \(\Gamma_ 2=0\), against \(H:\Gamma_ 1=0\), \(\Gamma_ 2\neq 0\), where \(\Gamma =\Sigma^{-1}\mu\) is Fisher's discriminant coefficient vector. Let \[ S=\sum_{\alpha}(X_{\alpha}-\bar X)(X_{\alpha}-\bar X)^ t, \quad Y=\sqrt{N}\bar X, \] \[ V=\sum_{\alpha}(W_{\alpha}-\bar W)(W_{\alpha}-\bar W)^ t, \quad Z=\sqrt{M}\bar W, \] \[ K=\sqrt{M/N}, \quad \nu =\sqrt{N}\mu, \quad U=S+YY^ t,\quad R=W+ZZ^ t. \] When needed, matrices and vectors are regarded as partitioned to conform with \(\mu\) and \(\Sigma\). A test is derived that rejects for \((Y_ 2+KZ)^ t(U_{22}+R)^{-1}(Y_ 2+KZ)>C_{\alpha}\), where \(C_{\alpha}\) is chosen to obtain the level of significance \(\alpha\). It is shown that the test is by likelihood ratio, that it is the uniformly most powerful invariant among a class of level \(\alpha\) invariant tests, that it is (with adequate adaptations) locally and asymptotically minimax, and that it is not type D in general [cf. \textit{S. L. Isaacson}, Ann. Math. Stat. 22, 217-234 (1951; Zbl 0043.345)]. The paper then studies the properties of this test for \(H_ 0:\mu_ 1=0\), \(\mu_ 2=0\) against \(H:\mu_ 1=0\), \(\mu_ 2\neq 0\), along similar lines. References to related problems in the literature are given.