This article is cited in
3 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
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.
UDC:
519.2 Received: 29.01.2003
DOI:
10.4213/dm202
Fulltext:
PDF file (681 kB)