An asymptotic likelihood ratio test for a markov chain with bernoullian or contingent input

A Markov chain with transition probabilitiesp _ij () that are functions of a parameter vector is defined. One oft input values is delivered to the chain on each trial. Under the hypothesis H₀ the parameter vector is independent of the input; under the hypothesis H₁ the vector is in general different for each different input. A likelihood ratio test for a single observation on a chain of great length is given for testing H₀ against H₁, given that the distribution of the inputs depends at most on the previous input and the present state of the chain. The test is therefore one of stationarity of the transition probabilities against a specific alternative form of nonstationarity. Application of the approach to a statistical test of lumpability of the states of a chain is indicated. Tests for other related hypotheses are suggested. Application of the test to within-subject effects for individual subjects is considered. Finally, some applications of the results in psychophysical contexts are suggested.Part of the final version of this paper was written at Stanford University during work sessions of a 1967 Summer Conference on Mathematical Models for Perception and Learning. Support was provided by the National Science Foundation through the Committee on Mathematics in the Social Sciences of the Institute for Advanced Study in the Behavioral Sciences.