Testing composite hypotheses about discrete-valued stationary processes
Given a discrete-valued sample X1; : : : ;Xn we wish to test whether it was generated by a stationary ergodic process belonging to a family H0, or it was generated by a stationary ergodic process outside H0. Apart from the assumptions of stationarity and ergodicity, no further probabilistic or parametric assumptions are made. We require the Type I error of the test to be uniformly bounded, while the probability of Type II error has to tend to zero as the sample size increases. For this notion of consistency we provide necessary and sufficient conditions on the family H0 for the existence of a consistent test. This criterion is illustrated with applications to testing for a membership to parametric families, generalizing some existing results.