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

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

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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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
2. M. I. Tikhomirova, V. P. Chistyakov, “Moving chi-square”, Discrete Math. Appl., 10:5 (2000), 469–475
3. V. M. Kruglov, “The Asymptotic Behavior of the Pearson Statistic”, Theory Probab Appl, 45:1 (2001), 69
4. B. I. Selivanov, “A family of multivariate chi-square statistics”, Discrete Math. Appl., 12:4 (2002), 401–413
5. B. I. Selivanov, “A family of multivariate $\chi^2$-statistics”, Discrete Math. Appl., 14:5 (2004), 527–533
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
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