Statistical inference for discrete patterns in dependent sequences

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We have studied the distributions of numbers of occurrences of discrete patterns and waiting time distributions for them in various dependent sequences. The following results have been derived.1. By using a combinatorial method it is shown that for every finite pattern, the distribution of the waiting time for the reversed pattern coincides with that of the waiting time for the original pattern in a multi-state dependent sequence with a certain type of exchangeability. The corresponding results for the waiting time for the r-th occurrence of the pattern, and for the number of occurrences of a specified pattern in n trials are also studied. Illustrative examples based on urn models are also given.2. Several waiting time random variables for a duplication within a memory window of size k in a sequence of {1,2,...,m}-valued random variables are investigated. The exact distributions of the waiting time random variables are derived by the method of conditional probability generating functio … More ns. In particular, the exact distribution of the waiting time for the first k-match is obtained when the underlying sequence is generated by higher order Markov dependent trials. Examples for numerical calculations are also given.3. We consider waiting time problems for two-dimensional pattern in a sequence of i.i.d. random vectors each of whose entries is 0 or 1. A general method for obtaining the exact distribution of the waiting time for the first occurrence of the pattern in the sequence is presented. The method is an extension of the method of conditional probability generating functions and it is very suitable for computations with computer algebra systems as well as usual numerical computations. Computational results applied to computation of exact system reliability are also given.4. Let k and m be positive integers with k>m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k-1 is obtained.5. Various new distributions related to runs and patterns in dependent sequences are obtained by using the stepwise smoothing formula for conditional expectations. Less