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Mat. Sb. (N.S.), 1969, Volume 79(121), Number 3(7), Pages 444–460 (Mi msb3598)  

This article is cited in 6 scientific papers (total in 6 papers)

Sequential $\chi^2$ criteria

V. K. Zakharov, O. V. Sarmanov, B. A. Sevast'yanov


Abstract: Independent trials with $m$ outcomes are considered. Let the probability of the $j$th outcome be $p_j$ under the null hypothesis $H$ but be $\widetilde p_j$ under the alternative hypothesis $\widetilde H$, $j=1,2,…,m$. For testing the hypothesis $H$ samples with increasing size $n_1<n_2<…<n_r$ are formed. We denote the number of times that the $j$th outcome appears in the first $n_i$ trials by $\nu_{ij}$. The statistics $\chi_i^2$ are introduced by formula (1.2). The hypothesis $H$ is rejected if $\chi_i^2>x_i^*$ for all $i=1,2,…,r$, where $x_i^*$ is some critical value, and is accepted in the remaining cases. The limit, for $n_i\to\infty$, of the distribution of $\chi^2$ under the hypotheses $H$ and $\widetilde H$ is given in the paper. These are used for the computation of the errors of the first and second kind, $\alpha$ and $\beta$, according to formulas (1.4) and (1.5). These distributions are multivariate generalizations of the central and noncentral $\chi^2$-distributions.
Bibliography: 4 titles.

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English version:
Mathematics of the USSR-Sbornik, 1969, 8:3, 419–435

Bibliographic databases:

UDC: 519.2
MSC: 62H15, 62H10, 62L10, 62M02, 60Exx
Received: 09.01.1969

Citation: V. K. Zakharov, O. V. Sarmanov, B. A. Sevast'yanov, “Sequential $\chi^2$ criteria”, Mat. Sb. (N.S.), 79(121):3(7) (1969), 444–460; Math. USSR-Sb., 8:3 (1969), 419–435

Citation in format AMSBIB
\Bibitem{ZakSarSev69}
\by V.~K.~Zakharov, O.~V.~Sarmanov, B.~A.~Sevast'yanov
\paper Sequential~$\chi^2$ criteria
\jour Mat. Sb. (N.S.)
\yr 1969
\vol 79(121)
\issue 3(7)
\pages 444--460
\mathnet{http://mi.mathnet.ru/msb3598}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=261751}
\zmath{https://zbmath.org/?q=an:0216.48301}
\transl
\jour Math. USSR-Sb.
\yr 1969
\vol 8
\issue 3
\pages 419--435
\crossref{https://doi.org/10.1070/SM1969v008n03ABEH002040}


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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. V. M. Kruglov, “Complete convergence of the Pearson statistic”, Math. Notes, 66:4 (1999), 515–519  mathnet  crossref  crossref  mathscinet  isi
    2. M. I. Tikhomirova, V. P. Chistyakov, “Moving chi-square”, Discrete Math. Appl., 10:5 (2000), 469–475  mathnet  crossref  mathscinet  zmath
    3. V. M. Kruglov, “The Asymptotic Behavior of the Pearson Statistic”, Theory Probab Appl, 45:1 (2001), 69  mathnet  crossref  mathscinet  isi  elib
    4. B. I. Selivanov, “A family of multivariate chi-square statistics”, Discrete Math. Appl., 12:4 (2002), 401–413  mathnet  crossref  mathscinet  zmath
    5. B. I. Selivanov, “A family of multivariate $\chi^2$-statistics”, Discrete Math. Appl., 14:5 (2004), 527–533  mathnet  crossref  crossref  mathscinet  zmath
    6. A. M. Zubkov, M. P. Savelov, “Convergence of the sequence of the Pearson statistics values to the normalized square of the Bessel process”, Discrete Math. Appl., 27:6 (2017), 405–411  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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