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

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.