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On runs and longest run tests: A method of finite Markov chain imbedding. (English) Zbl 0881.62086
Summary: Under very general assumptions on Bernoulli trials, a simple numerical method based on the finite Markov chain imbedding technique is systematically developed to determine (a) the joint distribution of the number of success runs and the number of successes; (b) the conditional distribution of the number of success runs given the number of successes, that is, the exact distribution of the success runs test statistic; (c) the joint distribution of the length of the longest success run and the number of successes; and (d) the conditional distribution of the length of the longest success run given the number of successes, that is, the exact distribution of the longest success run test statistic.
The critical regions and powers of these two tests (success runs and longest success run) are derived under the null hypothesis of independence and identical distribution (iid) as well as under the alternative hypothesis of Markov dependence. The application of the success runs test is illustrated using real data sets from a multicenter risk factor study on asthmatics.

MSC:
62M02 Markov processes: hypothesis testing
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
65C99 Probabilistic methods, stochastic differential equations
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