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Besides the different approaches suggested in the literature, accurateestimation of the order of a Markov chain from a given symbol sequence is anopen issue, especially when the order is moderately large. Here, parametricsignificance tests of conditional mutual information (CMI) of increasing order$m$, $I_c(m)$, on a symbol sequence are conducted for increasing orders $m$ inorder to estimate the true order $L$ of the underlying Markov chain. CMI oforder $m$ is the mutual information of two variables in the Markov chain being$m$ time steps apart, conditioning on the intermediate variables of the chain.The null distribution of CMI is approximated with a normal and gammadistribution deriving analytic expressions of their parameters, and a gammadistribution deriving its parameters from the mean and variance of the normaldistribution. The accuracy of order estimation is assessed with the threeparametric tests, and the parametric tests are compared to the randomizationsignificance test and other known order estimation criteria using Monte Carlosimulations of Markov chains with different order $L$, length of symbolsequence $N$ and number of symbols $K$. The parametric test using the gammadistribution (with directly defined parameters) is consistently better than theother two parametric tests and matches well the performance of therandomization test. The tests are applied to genes and intergenic regions ofDNA sequences, and the estimated orders are interpreted in view of the resultsfrom the simulation study. The application shows the usefulness of theparametric gamma test for long symbol sequences where the randomization testbecomes prohibitively slow to compute.

Markov chain order estimation with parametric significance tests of conditional mutual information