Csiszar’s ϕ-divergences for testing the order in a Markov chain

Abstract

Assume that a sequence of observations

$$x_1 ,...,x_{n + r} $$

(1)

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 H _o that a chain of order r — 1 will be sufficient on the basis of the tools given by the Statistical Information Theory: ϕ—Divergences. More precisely, if

$$p_{\alpha _1 ,...,\alpha _r :\alpha _{r + 1} } $$

(2)

denotes the transition probability for a r ^th order Markov chain, the hypothesis to be tested is

$$H_0 :p_{\alpha _1 ,...,\alpha _r :\alpha _{r + 1} } = p_{\alpha _2 ,...,\alpha _r :\alpha _{r + 1} } ,a_i \in \left\{ {1,...,s} \right\}, i = 1,..., r + 1$$

(3)

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.