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Teor. Veroyatnost. i Primenen., 1967, Volume 12, Issue 1, Pages 96–111 (Mi tvp688)  

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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English version:
Theory of Probability and its Applications, 1967, 12:1, 90–103

Bibliographic databases:

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
\Bibitem{Chi67}
\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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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. N. Tyulyagin, “On asymptotic admissibility of goodness-of-fit tests”, Math. USSR-Izv., 20:3 (1983), 535–576  mathnet  crossref  mathscinet  zmath
    2. I. G. Zhurbenko, È. M. Kudlaev, “Clarification of the interaction effect in randomized experiments”, Russian Math. Surveys, 39:1 (1984), 1–43  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. 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  mathnet  elib
    4. A. V. Ivanov, “Asimptoticheski optimalnye kriterii v zadache razlicheniya parametricheskikh gipotez o raspredelenii sluchainogo vektora. I”, Matem. vopr. kriptogr., 6:3 (2015), 89–116  mathnet  crossref  mathscinet  elib
    5. A. V. Ivanov, “Asimptoticheski optimalnye kriterii v zadache razlicheniya parametricheskikh gipotez o raspredelenii sluchainogo vektora. II”, Matem. vopr. kriptogr., 6:4 (2015), 49–64  mathnet  crossref  mathscinet  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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