This article is cited in 2 scientific papers (total in 2 papers)
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
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,
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Discrete Mathematics and Applications, 2003, 13:3, 319–329
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
\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.
\jour Discrete Math. Appl.
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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
D. V. Shuvaev, “O podposledovatelnostyakh markovskikh posledovatelnostei”, Matem. vopr. kriptogr., 7:4 (2016), 133–142
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