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This article is cited in 5 scientific papers (total in 5 papers)
A Theorem on Admissible Tests and Its Application to an Asymptotical Problem of Testing Hypotheses
D. M. Chibisov Moscow
Abstract:
Let $X_1,…,X_n$ be independent observations of a real random variable having a density $f(x,\vartheta)$, $\vartheta$ being a parameter from $s$-dimensional Euclidean space $E^s$. The hypothesis $\vartheta=0$ is tested. A test is said to be asymptotically (as $n\to\infty$) optimal if it has asymptotically best average power with respect to a given family of probability measures on the space of values of normed parameter $\theta=\vartheta/\sqrt n$ (see (2.1), (2.2)). It is shown that the problem, of constructing an asymtotically optimal test can be reduced to that of constructing an optimal (in the corresponding sense) test for some family of normal distributions in $E^s$ that differ only in locations.
In the proof the following result is used. Let $Q_0$ be a distribution in $E^s$ and $\{Q_0\}$, $\theta\in\Theta\subset E^s$ be the exponential family such that $dQ_\theta/dQ_0=\mathbf C(\theta)\exp(\theta,y)$, $y\in E^s$, $(\theta,y)$ denoting the scalar product. The hypothesis $\theta=0$ is tested. Then the following condition is necessary for a test $\varphi$ to be admissible: there exists a closed convex set $C\subset E^s$ such that (up to a set of $Q_0$-measure zero) $\varphi(y)=1$ if $y\in E^s\setminus C$ and $\varphi(y)=0$ if $y\in C^0$ where $C^0$ is the set of inner points of $C$.
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Theory of Probability and its Applications, 1967, 12:1, 90–103
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Received: 29.08.1966
Citation:
D. M. Chibisov, “A Theorem on Admissible Tests and Its Application to an Asymptotical Problem of Testing Hypotheses”, Teor. Veroyatnost. i Primenen., 12:1 (1967), 96–111; Theory Probab. Appl., 12:1 (1967), 90–103
Citation in format AMSBIB
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\by D.~M.~Chibisov
\paper A~Theorem on Admissible Tests and Its Application to an Asymptotical Problem of Testing Hypotheses
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 1
\pages 96--111
\mathnet{http://mi.mathnet.ru/tvp688}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=215402}
\zmath{https://zbmath.org/?q=an:0214.45802}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 1
\pages 90--103
\crossref{https://doi.org/10.1137/1112009}
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http://mi.mathnet.ru/eng/tvp688 http://mi.mathnet.ru/eng/tvp/v12/i1/p96
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A. N. Tyulyagin, “On asymptotic admissibility of goodness-of-fit tests”, Math. USSR-Izv., 20:3 (1983), 535–576
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I. G. Zhurbenko, È. M. Kudlaev, “Clarification of the interaction effect in randomized experiments”, Russian Math. Surveys, 39:1 (1984), 1–43
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V. M. Deundyak, Yu. V. Kosolapov, “O stoikosti kodovogo zashumleniya k statisticheskomu analizu nablyudaemykh dannykh mnogokratnogo povtoreniya”, Model. i analiz inform. sistem, 19:4 (2012), 110–127
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A. V. Ivanov, “Asimptoticheski optimalnye kriterii v zadache razlicheniya parametricheskikh gipotez o raspredelenii sluchainogo vektora. I”, Matem. vopr. kriptogr., 6:3 (2015), 89–116
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A. V. Ivanov, “Asimptoticheski optimalnye kriterii v zadache razlicheniya parametricheskikh gipotez o raspredelenii sluchainogo vektora. II”, Matem. vopr. kriptogr., 6:4 (2015), 49–64
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