Approximately minimax detecting of a vector signal in Gaussian noise

Yu. V. Linnik

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 4, pp. 561-578

Résumé

In a normal vector sample $(X_1,\dots,X_N)^T$ of independent identically distributed variables $X_i\in\mathscr N(\xi,\Sigma)$, the сovarianсe matrix $\Sigma$ is not supposed to be known, and the hypothesis $H_0$: $\xi=0$ against $H_1$: $N\xi^T\Sigma^{-1}\xi=\delta$ is tested. The Hotelling test $$ \Phi_N^0\colon T^2=N(N-1)X^TS^{-1}X>T_\varepsilon^2 $$ where $$ \overline X=N^{-1}\sum_{i=1}^NX_i;\quad S=\sum_{i=1}^N(X_i-X)(X_i-X)^T $$ is proved to be approximately minimax for large samples in the following sense: for all (randomized) tests $\Phi$ of level $\alpha=\alpha_N$ under conditions $$ O(\exp[-(\ln N)^{1/6}])\le\alpha\le1-O(\exp[-(\ln N)^{1/6}]) $$ and $\delta$'s under condition $$ \exp[-(\ln N)^{1/6}]\le\delta\le(\ln N)^{1/6} $$ we have $$ \sup_\Phi\inf_{\theta\in H_1}\mathbf E_\theta\Phi-\inf_{\theta\in H_1}\mathbf E_\theta\Phi_N^0=O_\varepsilon\biggl(\frac1{N^{i-\varepsilon}}\biggr) $$ for any $\varepsilon>0$.