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 Diskr. Mat., 2003, Volume 15, Issue 2, Pages 149–159 (Mi dm202)

On two chi-square-type statistics constructed from the frequencies of tuples of states of a multiple Markov chain

M. I. Tikhomirova, V. P. Chistyakov

Abstract: We consider a tuple of states of an $(s-1)$-order Markov chain whose transition probabilities depend on a small part of $s-1$ preceding states. We obtain limit distributions of certain $\chi^2$-statistics $X$ and $Y$ based on frequencies of tuples of states of the Markov chain. For the statistic $X$, frequencies of tuples of only those states are used on which the transition probabilities depend, and for the statistic $Y$, frequencies of $s$-tuples without gaps. The statistical test with statistic $X$ which distinguishes the hypotheses $H_1$ (a high-order Markov chain) and $H_0$ (an independent equiprobable sequence) appears to be more powerful than the test with statistic $Y$. The statistic $Z$ of the Neyman–Pearson test, as well as $X$, depends only on frequencies of tuples with gaps. The statistics $X$ and $Y$ are calculated without use of distribution parameters under the hypothesis $H_1$, and their probabilities of errors of the first and second kinds depend only on the non-centrality parameter, which is a function of transition probabilities. Thus, for these statistics the hypothesis $H_1$ can be considered as composite.
This research was supported by the Russian Foundation for Basic Research, grant 00–15–96136.

DOI: https://doi.org/10.4213/dm202

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English version:
Discrete Mathematics and Applications, 2003, 13:3, 319–329

Bibliographic databases:

UDC: 519.2

Citation: M. I. Tikhomirova, V. P. Chistyakov, “On two chi-square-type statistics constructed from the frequencies of tuples of states of a multiple Markov chain”, Diskr. Mat., 15:2 (2003), 149–159; Discrete Math. Appl., 13:3 (2003), 319–329

Citation in format AMSBIB
\Bibitem{TikChi03} \by M.~I.~Tikhomirova, V.~P.~Chistyakov \paper On two chi-square-type statistics constructed from the frequencies of tuples of states of a multiple Markov chain \jour Diskr. Mat. \yr 2003 \vol 15 \issue 2 \pages 149--159 \mathnet{http://mi.mathnet.ru/dm202} \crossref{https://doi.org/10.4213/dm202} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2006684} \zmath{https://zbmath.org/?q=an:1046.62080} \transl \jour Discrete Math. Appl. \yr 2003 \vol 13 \issue 3 \pages 319--329 \crossref{https://doi.org/10.1515/156939203322385928} 

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• https://doi.org/10.4213/dm202
• http://mi.mathnet.ru/eng/dm/v15/i2/p149

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This publication is cited in the following articles:
1. Yu. S. Kharin, A. I. Petlitskii, “A Markov chain of order $s$ with $r$ partial connections and statistical inference on its parameters”, Discrete Math. Appl., 17:3 (2007), 295–317
2. D. V. Shuvaev, “O podposledovatelnostyakh markovskikh posledovatelnostei”, Matem. vopr. kriptogr., 7:4 (2016), 133–142
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