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Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 4, Pages 561–578 (Mi tvp660)  

Approximately minimax detecting of a vector signal in Gaussian noise

Yu. V. Linnik

Leningrad

Abstract: In a normal vector sample $(X_1,…,X_N)^T$ 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
$$ \sup_\Phi\inf_{\theta\in H_1}\mathbf E_\theta\Phi-\inf_{\theta\in H_1}\mathbf E_\theta\Phi_N^0=O_\varepsilon(\frac1{N^{i-\varepsilon}}) $$
for any $\varepsilon>0$.

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English version:
Theory of Probability and its Applications, 1966, 11:4, 497–512

Bibliographic databases:

Received: 26.04.1966

Citation: Yu. V. Linnik, “Approximately minimax detecting of a vector signal in Gaussian noise”, Teor. Veroyatnost. i Primenen., 11:4 (1966), 561–578; Theory Probab. Appl., 11:4 (1966), 497–512

Citation in format AMSBIB
\Bibitem{Lin66}
\by Yu.~V.~Linnik
\paper Approximately minimax detecting of a~vector signal in Gaussian noise
\jour Teor. Veroyatnost. i Primenen.
\yr 1966
\vol 11
\issue 4
\pages 561--578
\mathnet{http://mi.mathnet.ru/tvp660}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=208773}
\zmath{https://zbmath.org/?q=an:0163.40204}
\transl
\jour Theory Probab. Appl.
\yr 1966
\vol 11
\issue 4
\pages 497--512
\crossref{https://doi.org/10.1137/1111058}


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