Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 401-417
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Yu. V. Linnik; Yu. V. Prokhorov; O. V. Shalaevskii. A contribution to the theory of aproximately mindmax detecting of a vector signal in a Gaussian noise. Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 401-417. http://geodesic.mathdoc.fr/item/TVP_1967_12_3_a0/
@article{TVP_1967_12_3_a0,
author = {Yu. V. Linnik and Yu. V. Prokhorov and O. V. Shalaevskii},
title = {A~contribution to the theory of aproximately mindmax detecting of a~vector signal in {a~Gaussian} noise},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {401--417},
year = {1967},
volume = {12},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1967_12_3_a0/}
}
TY - JOUR
AU - Yu. V. Linnik
AU - Yu. V. Prokhorov
AU - O. V. Shalaevskii
TI - A contribution to the theory of aproximately mindmax detecting of a vector signal in a Gaussian noise
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1967
SP - 401
EP - 417
VL - 12
IS - 3
UR - http://geodesic.mathdoc.fr/item/TVP_1967_12_3_a0/
LA - ru
ID - TVP_1967_12_3_a0
ER -
%0 Journal Article
%A Yu. V. Linnik
%A Yu. V. Prokhorov
%A O. V. Shalaevskii
%T A contribution to the theory of aproximately mindmax detecting of a vector signal in a Gaussian noise
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1967
%P 401-417
%V 12
%N 3
%U http://geodesic.mathdoc.fr/item/TVP_1967_12_3_a0/
%G ru
%F TVP_1967_12_3_a0
In a normal size $N$ sample of independent indentically distributed variables $X_i\in\mathbf N(\xi,\Sigma)$ with covariance matrix unknown, the hypothesis $H_0\colon\xi=0$ againts $H_\delta\colon N\xi^T\Sigma^{-1}\xi\ne\delta$ is tested. The well known Hotelling test $(T^2)$ is proved to be approximately minimax with considerable accuracy in the following sense: for all randomized level $\alpha$ tests $\Phi$ and a fixed positive $\delta$ we have: $$ \sup_\Phi\inf_{\theta\in H_\delta}\mathbf E_\theta\Phi-\inf_{\theta\in H_\delta}\mathbf E_\theta(T^2)=O(\exp c_1N^{1/4}\ln N) $$ with $\theta=(\xi,\Sigma),$$c_1>0$.
