Csiszar's phi-divergences for testing the order in a Markov chain

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Assume that a sequence of observations x(1),...,x(n+r) can be treated as the sample values of a Markov chain of order r or less (chain in which the dependence extends over r+1 consecutive variables only), and consider the problem of testing the hypothesis No that a chain of order r - 1 will be sufficient on the basis of the tools given by the Statistical Information Theory: rho -Divergences. More precisely, if p(a1),...,(ar:ar+1) denotes the transition probability for a r(th) order Markov chain, the hypothesis to be tested is H-0 : p(a1),...,(ar:ar+1) = p(a2),...,(ar):(ar+1), a(i) is an element of {1,...,s}, i = 1,..., r + 1 The tests given in this paper, for the first time, will have as a particular case the likelihood ratio test and the test based on the chi-squared statistic.
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