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Issue LXVI - N. 2 (2008)
Distributions of scan statistics in a sequence of Markov Bernoulli trials
We consider the binomial scan statistics (i.e. the number of scanning windows of length k containing specified number of successes where the window moves along the sequence of n Markov Bernoulli trials) N(1) n,k,r , N(2) n,k,r defined under the non-overlapping counting scheme, Mn,k,r , Mn,k,l,r and En,k,r under overlapping, l-overlapping and exactly r successes type of counting schemes respectively. In this paper we obtain the pgfs of distributions of above all binomial scan statistics and distribution of M = (Mn,k,1, Mn,k,2, . . . , Mn,k,k) using a unified approach based on the conditional probability generating functions. Distribution of M studied here is new in the literature. Further the distribution of scan statistics Ln,k (i.e. the maximum number of successes contained in a scanning window of length k moving along a sequence of n Markov Bernoulli trials) is obtained using the distribution of M. We give an algorithm to obtain the exact distributions of all scan statistics under consideration from the pgfs derived. We also study the waiting time random variables related to the binomial scan statistics.
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