The sequential chi-square test based on s-tuples of states of a Markov chain

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The detection of dependences in random sequences in the cases, where the dependence makes itself evident only on frequencies of long tuples of outcomes, involves difficulties connected with bounded possibilities of computers. In this case, the use is made of different criteria based on statistics depending on long tuples of successive symbols in the sequence under consideration or on incomplete tuples of symbols separated from each other. The criteria of such type can be found in [1]. In this paper, we find the limit distributions of chi-square statistics based on incomplete tuples of outcomes of independent trials in the case, where the number of trials tends to infinity, the outcomes are equiprobable, and the number of outcomes is finite.

Section: Covariances Of Normalised Frequenciesmentioning

confidence: 99%

A chi-square statistic based on the chains of outcomes of independent trials with gaps

Chistyakov¹

2001

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“…. , m, is asymptotically normal, which implies the joint asymptotic normality of both (7) and (5), and guarantee the convergence in distribution as n\ -> <*> of the random vector % 2 (n r ) determined by (6) to some r-dimensional random vector X = (Xl> · · · ?Xr)· We set ξί = (ξπ,··· ,ξ/m), ξ/ = (ξπ,..· ,ξι/π), 1=1,..., Γ, and The distributions of the random vectors ξ and ξ coincide with the corresponding limiting normal distributions of (7) and (5). In particular, the vector ξ has the rm-dimensional normal distribution with zero mean vector and the covariance matrix determined by formulas (2.4) in [1].…”

Section: Preliminary Wordsmentioning

confidence: 92%

Multivariate chi-square distribution for non-homogeneous polynomial scheme

Selivanov¹,

Chistyakov²

1998

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We consider a statistic whose components are the χ 2 statistics, constructed on the base of the frequencies of outcomes of non-homogeneous polynomial samples of growing volumes. Conditions are formulated which guarantee the existence of a limiting distribution of this statistics, and its Laplace transform is presented. In the homogeneous case, the Laplace transform obtained is reduced to the form which has been known before.This work was supported by the Russian Foundation for Basic Research, grants 96-01-00531, 96-15-96092. FORMULATION OF THE PROBLEMIn [1], a multivariate chi-square distribution was obtained, which was a limit distribution of the set of chi-square statistics based on frequencies of outcomes of growing polynomial samples. The distribution obtained was used in [1] to calculate the error probabilities of the r-fold chi-square test of sequential type.The results of [1] are cited in [2], where other approaches to the definition of multivariate chi-square distributions were presented. Some generalizations and applications of the r-fold chi-square test were given in [3][4][5].In the change-point problem, to evaluate the characteristics of tests used, we need to know the distribution of the corresponding statistics in the case of non-homogeneous polynomial scheme. In this paper, we formulate the conditions which guarantee the existence of a limiting distribution of the set of chi-square statistics in that case, and give its Laplace transform.We consider a sequence of independent (non-homogeneous polynomial) trials with m outcomes 1 , 2, . . . , m. Let p\ (t) , . . . , p m (t) be the probabilities of these outcomes in the ith trial, t = 1 , 2, . . . We fix r, r > 2, natural numbers 1 < n\ < . . . < n r , and assume that for n\ -> <»(1)

“…Let f¼ ® g, ® D .® 1 ; : : : ® s /, stand for the stationary distributionof Markov chain (3). We assume that Pfx 1 ; D ® 1 ; x 2 D ® 2 ; : : : ; x s D ® s g D ¼ ® :…”

Section: A High-order Markov Chainunclassified

“…Let ¹ ® .n/ denote the frequency of the state ® in n leading terms of sequence (3), and let º i .n/ denote the frequency of the tuple i D .i 1 ; i 2 ; i 3 / in n leading terms of the sequence of 3-tuples…”

Section: A High-order Markov Chainmentioning

confidence: 99%

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On two statistics of chi-square type based on frequencies of tuples of states of a high-order Markov chain

Тихомирова¹,

Chistyakov²

2003

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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 Â 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 NeymanPearson 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 rst 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.