Discriminating between sequences of bernoulli and markov-bernoulli trials

Abstract

Givenp↦ (0,1), we consider a sequence of {0,1}-valued random variables (a) that have an i.i.d. Bernoulli (p) distribution or (b) which evolve according to a stationary ergodic 2-state Markov chain with transition probabilities given by , and with stationary distribution . Lehmann (1986) proved that the conditional run test possesses certain optimality properties if used as a criterion to discriminate between the above two possibilities; we show that the same is true of the unconditional analog of the run test if p is small and n is not excessively large. The rather complicated distribution of R, the total number of runs of successes or failures, is approximated (in the total variation sense) by an appropriately defined Poisson distribution on the odd integers, with the approximation performing well for low values of p - independently of the value of n. The test based on the above approximation is shown to be"almost"consistent Our procedure is generalized to test the hypothesis of independence against the alternative of second-order Markov dependence; the case of r th-order Markov dependence (r≥3) can be handled in much the same way.