Locally and Asymptotically Minimax Tests of a Multivariate Problem

Abstract

Let $X_1, \cdots, X_N$ be independent normal $p$-vectors with common mean vector $\xi = (\xi_1, \cdots, \xi_p)'$ and common non-singular covariance matrix $\Sigma$. Write $N\bar{X} = \sum^N_1 X_i, S = \sum^N_1 (X_i - \bar{X})(X_i - \bar{X})', b\lbrack i\rbrack$ for the $i$-vector consisting of the first $i$ components of a $p$-vector $b$ and $C\lbrack i\rbrack$ for the upper left-hand $i \times i$ sub-matrix of a $p \times p$ matrix $C$. Let $\delta = N\xi'\Sigma^{-1}\xi (\geqq0)$. We will consider here the problem of testing the hypothesis $H_0:\xi_1 = \cdots = \xi_p = 0$ against the alternative $H_\lambda:\xi_1 = \cdots = \xi_q = 0,\quad \delta = \lambda,$ where $\xi, \Sigma$ are unknown, $q < p$ and $\lambda > 0$ is given. The problem of testing $H_0$ against $H_\lambda$ remains invariant under the group $G$ of $p \times p$ non-singular matrices \begin{equation*}\tag{1.0}g = \begin{pmatrix}g_{11} 0 \\ g_{21} g_{22}\end{pmatrix}\end{equation*} where $g_{11}$ is a $q \times q$ sub-matrix of $g$. A maximal invariant in the space of $(\bar{X}, S)$ under $G$ is $\bar{R} = (\bar{R}_1, \bar{R}_2)$ where \begin{equation*}\tag{1.1}\bar{R}_1 + \bar{R}_2 = N\bar{X}'(S + N\bar{X}\bar{X}')^{-1}\bar{X}\end{equation*}\begin{equation*}\bar{R}_1 = N\bar{X}'_{\lbrack q\rbrack}(S_{\lbrack q\rbrack} + N\bar{X}_{\lbrack q\rbrack}\bar{X}'_{\lbrack q\rbrack})^{-1}\bar{X}_{\lbrack q\rbrack};\end{equation*} and a corresponding maximal invariant in the parametric space of $(\xi, \Sigma)$ under $G$ is $\bar{\Delta} = (\bar\delta_1, \bar\delta_2)$, where \begin{equation*}\tag{1.2}\bar\delta_1 + \bar\delta_2 = N\xi'\Sigma^{-1}\xi,\end{equation*}\begin{equation*}\bar\delta_1 = N\xi'_{\lbrack q\rbrack}\Sigma^{-1}_{\lbrack q\rbrack}\xi_{\lbrack q\rbrack}\end{equation*} (see Giri (1961)). Giri (1962) has shown that for testing $H_0$ against the alternatives \begin{equation*}\tag{1.3}H_1':\xi_1 = \cdots = \xi_q = 0,\quad\delta > 0,\end{equation*} the likelihood ratio test of significance level $\alpha$ is given by \begin{align*}\tag{1.4}\phi(X_1, \cdots, X_N) = 1, \quad \text{if} \quad Z = (1 - \bar{R}_1 - \bar{R}_2)/(1 - \bar{R}_1) \leqq C, \\ = 0, \quad \text{otherwise};\end{align*} where the constant $C$ is chosen in such a way that $E_{H_0}\phi = \alpha$ and under $H_0, Z$ has central beta distribution with parameters $(N - p)/2, (p - q)/2$. It follows from Giri (1961) (also from (2.3) in Section 2) that with respect to $G$ the likelihood ratio test for testing $H_0$ against $H'_1$ is not uniformly most powerful invariant and that there is no uniformly most powerful invariant test for this problem. However, for fixed $p$, this test is nearly optimum as the sample size becomes large (Wald (1943)). Thus if $p$ is not large, it seems likely that the sample size occurring in practice will usually be large enough for this result to be relevant. However, if the dimension $p$ of the basic multivariate distribution is large, it may be that the sample size must be extremely large for this large sample result to apply, for example, there are cases where $N/p^3$ is large. The only satisfactory property of this test procedure known to us at this writing is that the difference of the powers of this test and the best invariant test with respect to $G$ is $O(N^{-1})$ when $p, q$ are both $O(N), \delta$ is $O(N^{\frac{1}{2}})$ and $N$ becomes large (Giri (1967)). In this paper it will be shown that the likelihood ratio test for this problem is neither locally minimax as $\lambda \rightarrow 0$ nor asymptotically minimax as $\lambda \rightarrow \infty$. Thus it will be established here that it can not be minimax for every $\lambda$ for this problem. Attempts are also made to find test procedures based on $\bar{R}$ which are locally and asymptotically minimax. It is easy to see that the power function of any test procedure based on $\bar{R}$ is constant on each contour $\delta = \lambda$. It will be shown in Sections 2 and 3 that the test (hereafter called $\phi^\ast$) which rejects $H_0$ if $\bar{R}_1 + (N - q)\bar{R}_2/(p - q) \geqq C_\alpha$, where $C_\alpha$ is chosen so that $E_{H_0}\phi^\ast = \alpha$, is locally minimax as $\lambda \rightarrow 0$ but not asymptotically logarithmically (sometimes will be called asymptotically only) minimax as $\lambda \rightarrow \infty$ and the test (hereafter called Hotelling's test) which rejects $H_0$ if $\bar{R}_1 + \bar{R}_2 \geqq C_\alpha'' (C_\alpha''$ depends on the size $\alpha$ of the test) is asymptotically minimax as $\lambda \rightarrow \infty$ but not locally minimax as $\lambda \rightarrow 0$. Furthermore for this problem no invariant test (under $G$) is minimax for every $\lambda$. This includes the likelihood ratio test, Hotelling's test and $\phi^\ast$. It may be pointed out here that none of these last two test procedures is derived following after any well known statistical theory. They are derived so as to possess the required local and asymptotic property stated above.