How to test the Markov property of a 0, 1 sequence? If a sequence of 0, 1 (such as 001010101101111......) is given, How can i test whether this sequenc can be treated as 1-order Markov sequence。
If possible, please give me the test method and an example. Thanks.
Dear Shaofeng
I think you can use independence tests such as gap, poker, difference sign, runs, rank tests. If independence reject in your data then you can use one-way chi-square test to compare probability of p00, p01, p10, p11 with your supposed probabilities in your transition matrix.
I think that a complementary comment will be useful. Namely, some more precise info about the test proposed by @Ette. Right now I am ready to present only a formal setting of the statistics :
Pkl := Nkl/Nk , where for given sequence Xi, i=1,2,...,n consisting of n entries equal 0 or 1 only,
1. we assume N0 or N1 equal the number of 0-s and 1-s among first n-1 elements Xi
2. we build a sequence of n-1 pairs Yi = (Xi,Xi+1), i=1,2,...,n-1;
3. for every possible values of the pairs (k,l) = (0,0), (0,1),(1,0), (1,1) we assume Nkl equal the number of indices i such that Yi=(k,l).
The pairs are not iid, and therefore the test statistics Pkl may follow a more compilcated distribution than just the usual binomial one (as it would be for independent testing of each transition pds - examined for each pair (k,l) separately), Therefore I suggest to seek formulas for the joint pd of the four estimators Pkl, with (k,l)= (0,0), (0,1),(1,0), (1,1). After a while I should have some idea about the limit pd as n tends to infinity. Seemingly this should be a normal pd for the pair (P01,P11) with a covariance matrix dependent on the tested probabilities, with deterministically dependent P00=1 - P01 and P10= P11. Correspondingly, asymptotically some Pearson-like chi2 test should then follow, consistently with @Seyed's suggestion.
Best regards, Joachim
PS. Perhaps someone knows reference to such a test built on ONE sequence? This would allow me to get sleep earlier :)
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