Abstract
Consider a sequence of n independent Bernoulli trials with the j-th trial having probability pj of success, 1 ≤ j ≤ n. Let M(n,K) and N(n, K) denote, respectively, the r-dimensional random variables (M(n, k1),..., M(n,kr) and (N(n,k1), ..., N(n, kr)), where K = (k1, k2, ..., kr) and M(n, s) [N(n, s)] represents the number of overlapping [non-overlapping] success runs of length s. We obtain exact formulae and recursions for the probability distributions of M(n, K) and N(n, K). The techniques of proof employed include the inclusion-exclusion principle and generating function methodology. Our results have potential applications to statistical tests for randomness.
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REFERENCES
Agin, M. and Godbole, A. P. (1992). A new exact runs test for randomness, Computing Science and Statistics: Proceedings of the 22nd Symposium on the Interface (eds. C. Page and R. LePage), 281–285, Springer, New York.
Barbour, A., Holst, L. and Janson, S. (1992). Poisson Approximation, Oxford University Press, Oxford.
Chryssaphinou, O. and Papastavridis, S. (1988). A limit theorem on the number of overlapping occurrences of a pattern in a sequence of independent trials, Probab. Theory Related Fields, 79, 129–143.
Chryssaphinou, O., Papastavridis, S. and Tsapelas, T. (1993). On the number of overlapping success runs in a sequence of independent Bernoulli trials, Applications of Fibonacci Numbers, Vol. 5: Proceedings of the 5th International Conference on Fibonacci Numbers (eds. G. E. Bergum, A. Horadam and A. Philippou), 103–112, Kluwer, Boston.
Godbole, A. P. (1990). Specific formulæ for some success run distributions, Statist. Probab. Lett., 10, 119–124.
Godbole, A. P. (1991). Poisson approximations for runs and patterns of rare events, Adv. in Appl. Probab., 23, 851–865.
Godbole, A. P. (1992a). Approximate reliabilities of m-consecutive-k-out-of-n: failure systems, Statist. Sinica, 3, 321–328.
Godbole, A. P. (1992b). The exact and asymptotic distribution of overlapping success runs, Comm. Statist. Theory Methods, 21, 953–967.
Hirano, K. (1986). Some properties of the distributions of order k, Fibonacci Numbers with Applications: Proceedings of the 1st International Conference on Fibonacci Numbers and their Applications (eds. A. Philippou, G. Bergum and A. Horadam), Reidel, Dordrecht.
Koutras, M. and Papastavridis, S. (1992). On the number of runs and related statistics, Statist. Sinica, 3, 277–294.
Ling, K. D. (1988). On binomial distributions of order k, Statist. Probab. Lett., 6, 247–250.
Panaretos, J. and Xekalaki, E. (1986). On some distributions arising from certain generalized sampling schemes, Comm. Statist. Theory Methods, 15, 873–891.
Panarctos, J. and Xekalaki, E. (1989). A probability distribution associated with events with multiple occurrences, Statist. Probab. Lett., 8, 389–395.
Papastavridis, S. (1990). m-consecutive-k-out-of-n: F systems, IEEE Translation on Reliability, R39, 386–388.
Philippou, A. N. and Makri, F. (1986). Successes, runs and longest runs, Statist. Probab. Lett., 4, 211–215.
Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties, Statist. Probab. Lett., 1, 171–175.
Philippou A. N., Tripsianis, G., and Antzoulakos, D. (1989). New Polya and Inverse Polya Distributions of order k, Comm. Statist. Theory Methods, 18, 2125–2137.
Xekalaki, E., Panaretos, J. and Philippou, A. (1987). On some mixtures of distributions of order k, Fibonacci Quarterly, 25, 151–160.
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Godbole, A.P., Papastavridis, S.G. & Weishaar, R.S. Formulae and Recursions for the Joint Distribution of Success Runs of Several Lengths. Annals of the Institute of Statistical Mathematics 49, 141–153 (1997). https://doi.org/10.1023/A:1003170823986
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DOI: https://doi.org/10.1023/A:1003170823986