On Some Statistical Tests for $M$th Order Markov Chains

Abstract

Certain $\psi^2$ statistics were defined by Good in [8] and he stated that there is a "strong analogy" between certain functions of these statistics and some given likelihood ratio (LR) statistics appropriate for testing hypotheses concerning the order of a Markov chain. He also indicated that the analogy held when the hypothesis of "perfect randomness" is true. The present author has indicated in [12] that, for the $\psi^2$ statistics in [8], this analogy (i.e., the asymptotic equivalence of the corresponding statistics) does not hold under some more general conditions when the "perfect randomness" hypothesis is not true. It will be seen herein that certain functions of a modified form of the $\psi^2$ statistics are asymptotically equivalent to certain LR statistics in the more general case when the hypothesis $H(P_m)$ that the positively regular Markov chain (see [2]) is governed by a completely specified system $P_m$ of $m$th order transition probabilities is true. Also, certain functions of a different modified form of the $\psi^2$ statistics will be seen to be asymptotically equivalent to certain LR statistics in the case when the hypothesis $H_m$ that the positively regular Markov chain is of order $m$ is true. These results are helpful in determining the asymptotic distributions of various statistics and the null hypotheses that can be tested with a given statistic. For example, if a given statistic $G$ is asymptotically equivalent, under $H(P_1),$ to the LR statistic $L$ for testing the null hypothesis $H(P_1)$ within the alternate hypothesis $H_2$, then the asymptotic distribution, under $H(P_1),$ of $G$ will be $\chi^2$ with a known number of degrees of freedom (i.e., with a known expectation); $G$ can be used directly to test $H(P_1)$ within $H_2$ (if $G$ is sensitive to these hypotheses), although the asymptotic distribution, under $H_2,$ of $G$ may differ, in a certain sense, from that of $L$ (see Section 6 in [1]). However, if a given statistic $\Delta G$ is asymptotically equivalent, under $H(P_1),$ to the LR statistic $\Delta L$ for testing the null hypothesis $H_1$ within the alternate hypothesis $H_2$ (i.e., if there is an "ostensible analogy" between $\Delta G$ and $\Delta L$), but this asymptotic equivalence does not hold under some more general conditions (e.g., under $H_1$), then $\Delta G$ can not be used to test $H_1$ within $H_2$; the asymptotic distribution, under $H(P_1),$ of $\Delta G$ will be $\chi^2,$ but the asymptotic distribution, under the null hypothesis $H_1,$ will not be $\chi^2,$ and furthermore the expectation, under $H_1,$ of $G$ can approach infinity (see [15]). The present author has indicated in [15] that certain functions of a modified form of the $\psi^2$ statistics, which were investigated by Stepanow in [18] and which are computed for a specified $P_1$, are asymptotically equivalent, under $H(P_1),$ to certain LR statistics, but that they will not be equivalent under $H_1.$ Although it is stated in [18] that the results presented there can be applied to the solution of the problem of testing the null hypothesis $H_1,$ it is shown in [15] that none of the statistics in [18] can be used directly to test this composite hypothesis. In the present paper, it will be seen that a statistic based on a different modified form of $\psi^2,$ as well as certain other statistics described herein, will be asymptotically equivalent, under $H_1,$ to a certain LR statistic and can be used to test the null hypothesis $H_1.$ The $\psi^2$ statistics defined by Good in [10] are more general than those given in [3], [8], [18]. Besides studying the relation between these statistics and the LR statistics, we shall also discuss certain conjectures proposed in [10] concerning the asymptotic distributions of these statistics, which were investigated by Billingsley [4] for the cases $H_0$ and $H_1$ (the author mentions that a more general result for $H_m(m \geqq 0)$ can be obtained using similar methods) and independently, using different methods, by the present author [14] for the case $H_m(m \geqq 0)$ when the transition probabilities are all positive (this author also mentions that a more general result can be obtained by similar methods). The $\psi^2$ and LR statistics defined in [10] and [8], as well as some related statistics developed in the present paper, will be generalized further herein, and the asymptotic distributions of these generalized statistics will be investigated. This investigation leads to generalizations of the asymptotic distributions obtained by Good [8], Billingsley [3] [4], and the present author [13] [14], and it helps to clarify the relation between the various statistics. The different asymptotically equivalent forms of various statistics presented here make it possible for the statistician to choose whichever form he finds preferable both from the computational point of view and also from some other viewpoints (see [1], [5], [11]).