Abstract Generated abstract
This paper studies minimax properties of the usual squared multiple correlation test \(R^2\) for testing zero correlation in a multivariate normal sample, focusing on the case \(p=3\) with large sample size. Building on earlier work of Giri, Kiefer, Linnik, Prokhorov, and Shalaevskii, it proves that the test rejecting for \(R^2>C\) is \(\varepsilon\)-minimax for testing \(H_0:\rho^2=0\) against both the simple alternative \(\rho^2=\delta\) and the composite alternative \(\rho^2\ge\delta\). Under stated bounds on the level and separation parameter, the minimax deficiency is shown to be \(O(1/N^k)\) for any prescribed natural number \(k\). The proof relies on constructing an approximate solution to the Giri, Kiefer integral equation with uniformly small residual.
Full Text
UDC 519.24/27
MATHEMATICS
LI HOANG TU
AN APPROXIMATE MINIMAX PROPERTY OF THE TEST \(R^2\)
(Presented by Academician Yu. V. Linnik on 3 I 1968)
Let \(X_1,\ldots,X_N\) be independent \(p\)-dimensional vectors obeying the normal law \(N(\xi,\Sigma)\) (here, as usual, \(\xi=EX_i\), \(\Sigma=E(X_i-\xi)(X_i-\xi)^T\) is the vector of means and the correlation matrix; vectors are written as columns, \(T\) denotes transposition). Put
\[ N\overline X=\sum_{i=1}^{N}X_i,\qquad S=\sum_{i=1}^{N}(X_i-\overline X)(X_i-\overline X)^T, \]
\[ \Sigma= \begin{pmatrix} \Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22} \end{pmatrix}, \qquad S= \begin{pmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{pmatrix}. \]
(where \(\Sigma_{22}, S_{22}\) are \((p-1)\times(p-1)\)-matrices).
\[ \rho^2=\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}/\Sigma_{11}. \]
We shall test the statistical hypothesis \(H_0:\rho^2=0\) against the alternative \(H_1:\rho^2=\delta,\ 0<\delta<1\), and against the alternative \(H_1':\rho^2\ge \delta\), with level \(\alpha,\ 0<\alpha<1\).
To solve the two indicated problems one usually uses the \(R^2\)-test, which rejects the hypothesis \(H_0\) when \(R^2=S_{12}S_{22}^{-1}S_{21}/S_{11}>C\), where the constant \(C\) is determined by the level \(\alpha\).
In the case \(N\le p\), if \(\xi\) is unknown, or \(N\le p-1\), if \(\xi\) is known, the test \(R^2\) is equal to a constant (if it exists).
In 1964, H. Giri and J. Kiefer proved the minimax property of the test \(R^2\) for the case \(p=3,\ N=4\) (or \(N=3\), when \(\xi\) is known).
The study of samples of size \(N>4\) for \(p=3\) encounters considerable difficulties. However, for \(N\to\infty\), as in \((^3)\), we proved the \(\varepsilon\)-minimaxity of the test \(R^2\) for arbitrary values \(p\ge 3\) \((\varepsilon=O(1/N^{1-\varepsilon_1})\), where \(\varepsilon_1\) is any prescribed small positive number).
In the present paper we apply the methods used by Yu. V. Linnik, Yu. V. Prokhorov, O. V. Shalaevskii, H. Giri, and J. Kiefer in \((^{3,4,1})\), in order to prove \(\varepsilon\)-minimaxity for the test \(R^2\) when \(p=3,\ N\) is large. \((\varepsilon=O(1/N^k)\), where \(k\) is any prescribed natural number.)
Theorem 1. The test \(R^2:\ R^2>C\) for testing the hypothesis \(H_0\) against \(H_1\) is \(\varepsilon\)-minimax for any level \(\alpha\in(0,1)\).
For any \(\varepsilon>0\) the relation holds:
\[ \operatorname{Sup}_{\Phi}\inf_{\theta\in H_1}E_\theta\Phi - \inf_{\theta\in H_1}E_\theta\Phi_N \le \varepsilon \tag{1} \]
for \(N>N_0(\varepsilon)\). Here \(\theta=(\xi,\Sigma)\) is the parameter; \(\Phi_N\) is the test \(R^2\); \(\Phi\) ranges over all tests of level \(\le \alpha\).
More precisely: if the level \(\alpha=\alpha_N\) is subject to the conditions
\[ O(\lambda/N^k)\le \alpha \le \lambda-O(1/\ln N) \tag{2} \]
and
\[ 1/N\ln N\le \delta\le 1, \tag{3} \]
then
\[ \operatorname{Sup}_{\Phi}\inf_{\theta\in H_1} E_\theta\Phi-\inf_{\theta\in H_1}E_\theta\Phi_N =O(1/N^k). \tag{4} \]
When the hypothesis \(H_1\) is replaced by the hypothesis \(H_1'\), an analogous theorem holds:
Theorem 2. The test \(R^2\): \(R^2>C\) for testing \(H_0\) against the composite hypothesis \(H_1'\) is \(\varepsilon\)-minimax for any \(\alpha\in(0,1)\) and \(0<\delta<1\). Under the conditions of Theorem 1, relation (1) holds with \(H_1\) replaced by \(H_1'\).
To prove these theorems we establish a lemma asserting the existence of an approximate solution \(\lambda\) of the Giri–Kiefer integral equation.
This equation has the form (the case \(p=3\))
\[ \begin{aligned} &\int_{\Gamma} \left[ 1+\sum_{i=2}^{3} r_i\left(\frac{1-\delta}{\gamma_i}-1\right) \right]^{-N/2} \sum_{\beta_2=0}^{\infty}\sum_{\beta_3=0}^{\infty} \frac{\Gamma(N/2+\beta_2+\beta_3)}{\Gamma(N/2)} \times \\ &\quad \times \prod_{i=2}^{3} \left\{ \frac{\Gamma((N-i+2)/2+\beta_i)} {\Gamma((N-i+2)/2)(2\beta_i)!} \left( \frac{4r_i a_i^2} {1+\sum_{j=2}^{3} r_j\left(\frac{1-\delta}{\gamma_j}-1\right)} \right)^{\beta_i} \right\} \,d\lambda(\Delta) = \\ &\quad = F\left(\frac{N}{2},\frac{N}{2},1,C\delta\right), \end{aligned} \]
where \(\Gamma\) is the simplex \((\delta_2,\delta_3)\), \(\delta_i\ge 0\), \(\delta_2+\delta_3=\delta\); \((r_2,r_3)\) are parameters, \(r_2,r_3>0\), \(r_2+r_3=C\); \(\gamma_i=1-\sum_{j=2}^{i}\delta_j\), \(\gamma_1=1\), \(a_i^2=\delta_i\gamma_3/\gamma_{i-1}\gamma_i\); \(F\) is the hypergeometric function.
Lemma. As \(N\to\infty\) and \(1/N\ln N\le \delta\le 12K\ln n/N\), \(2/\ln N\le C\le 2K\ln n/N\), there exists a continuous probability distribution \(\lambda_{C\delta N}\) on \(\Gamma\) which, when substituted into (2), gives a residual, uniformly in \(r_2,r_3\ge 0\), \(r_2+r_3=C\), not exceeding \(O(1/N^k)\), where \(k\) is any fixed prescribed natural number.
In conclusion, I express my deep gratitude to Acad. Yu. V. Linnik, and also to O. V. Shalaevskii.
Leningrad State University
named after A. A. Zhdanov
Received
21 XII 1967
REFERENCES
- N. Giri, J. Kiefer, Ann. Math. Statist., 35, No. 4 (1964).
- Yu. V. Linnik, DAN, 169, No. 3 (1966).
- Yu. V. Linnik, Theory of Probability and Its Applications, 12, issue 4 (1966).
- Yu. V. Linnik, Yu. V. Prokhorov, O. V. Shalaevskii, Theory of Probability and Its Applications, 12, issue 3 (1967).