Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T18:08:11.747Z Has data issue: false hasContentIssue false
  • English
  • Français

Success run statistics defined on an urn model

Part of: Distribution theory Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Frosso S. Makri ,
Andreas N. Philippou  and
Zaharias M. Psillakis
Frosso S. Makri*
Affiliation:
University of Patras
Andreas N. Philippou*
Affiliation:
University of Patras
Zaharias M. Psillakis*
Affiliation:
University of Patras
*
∗ Postal address: Department of Mathematics, University of Patras, 26500 Patras, Greece.
∗ Postal address: Department of Mathematics, University of Patras, 26500 Patras, Greece.
∗∗∗∗ Postal address: Department of Physics, University of Patras, 26500 Patras, Greece. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

Keywords

Success runwaiting timeurn modelPólya-Eggenberger sampling schemelinear and circular binary sequencesBernoulli trialnonparametric test of randomnessreliability of consecutive systems

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 62E15: Exact distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 991 - 1019
Copyright
Copyright © Applied Probability Trust 2007 

References

Antzoulakos, D. L., Bersimis, S. and Koutras, M. V. (2003a). On the distribution of the total number of run lengths. Ann. Inst. Statist. Math. 55, 865–884.CrossRefGoogle Scholar
Antzoulakos, D. L., Bersimis, S. and Koutras, M. V. (2003b). Waiting times associated with the sum of success run lengths. In Mathematical and Statistical Methods in Reliability, eds Lindqvist, B. and Doksum, K., World Scientific, River Edge, NJ, pp. 141–157.Google Scholar
Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Barton, D. A. and David, F. N. (1958). Runs in a ring. Biometrika 45, 572–578.CrossRefGoogle Scholar
Burr, E. J. and Cane, G. (1961). Longest run of consecutive observations having a special attribute. Biometrica 48, 461–465.Google Scholar
Charalambides, C. A. (2002). Enumerative Combinatorics. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
Cochran, W. G. (1938). An extension of Gold's method of examining the apparent persistence of one type of weather. Quart. J. R. Meteor. Soc. 64, 631–634.Google Scholar
Eryilmaz, S. and Demir, S. (2007). Success runs in a sequence of exchangeable binary trials. J. Statist. Planning Inference 137, 2954–2963.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058.Google Scholar
Fu, J. C. and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Imbedding Approach. World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Fu, J. C., Lou, W. Y. W., Bai, Z. and Li, G. (2002). The exact and limiting distributions for the number of successes in success runs within a sequence of Markov-dependent two-state trials. Ann. Inst. Statist. Math. 54, 719–730.CrossRefGoogle Scholar
George, E. O. and Bowman, D. (1995). A full likelihood procedure for analysing exchangeable binary data. Biometrics 51, 512–523.CrossRefGoogle ScholarPubMed
Gibbons, J. D. and Chakraborti, S. (2003). Nonparametric Statistical Inference, 4th edn. Marcel Dekker, New York.Google Scholar
Godbole, A. P. (1990). On hypergeometric and related distributions of order k . Commun. Statist. Theory Meth. 21, 1291–1301.Google Scholar
Goldstein, L. (1990). Poisson approximation in DNA sequence matching. Commun. Statist. Theory Meth. 19, 4167–4179.Google Scholar
Hirano, K. and Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain. Statistica Sinica 3, 313–320.Google Scholar
Johnson, N. and Kotz, S. (1977). Urn models and Their Application. John Wiley, New York.Google Scholar
Koutras, M. V. and Alexandrou, V. A. (1997). Non-parametric randomness tests based on success runs of fixed length. Statist. Prob. Lett. 32, 393–404.Google Scholar
Koutras, M. V., Papadopoulos, G. K. and Papastavridis, S. G. (1995). Runs on a circle. J. Appl. Prob. 32, 396–404.Google Scholar
Ling, K. D. (1988). On discrete distributions of order k defined on Pólya–Eggenberger urn model. Soochow. J. Math. 2, 199–210.Google Scholar
Lou, W. Y. W. (2003). The exact distribution of the k-tuple statistic for sequence homology. Statist. Prob. Lett. 61, 51–59.Google Scholar
Makri, F. S. and Philippou, A. N. (1994). Binomial distributions of order k on the circle. In Runs and Patterns in Probability, eds Godbole, A. P. and Papastavridis, S. G., Kluwer, Dordrecht, pp. 65–81.Google Scholar
Makri, F. S., Philippou, A. N. and Psillakis, Z. M. (2007a). Pólya, inverse Pólya, and circular Pólya distributions of order k for l-overlapping success runs. Commun. Statist. Theory Meth. 36, 657–668.Google Scholar
Makri, F. S., Philippou, A. N. and Psillakis, Z. M. (2007b). Shortest and longest length of success runs in binary sequences. J. Statist. Planning Inference 137, 2226–2239.CrossRefGoogle Scholar
Mood, A. M. (1940). The distribution theory of runs. Ann. Math. Statist. 11, 367–392.Google Scholar
Móri, T. F. (1991). On the waiting time till each of some given patterns occurs as a run. Prob. Theory Relat. Fields 87, 313–323.Google Scholar
Mosteller, F. (1941). Note on an application of runs to quality control charts. Ann. Math. Statist. 12, 228–232.Google Scholar
Muselli, M. (1996). Simple expressions for success run distributions in Bernoulli trials. Statist. Prob. Lett. 31, 121–128.Google Scholar
Panaretos, J. and Xekalaki, E. (1986). On some distributions arising from certain generalized sampling schemes. Commun. Statist. Theory Meth. 15, 873–891.Google Scholar
Philippou, A. N. (1986). Distributions and Fibonacci polynomials of order k, longest runs, and reliability of consecutive-k-out-of-n:F systems. In Fibonacci Numbers and Their Applications, eds Philippou, A. N. et al., Reidel, Dordrecht, pp. 203–227.CrossRefGoogle Scholar
Philippou, A. N., Georgiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties. Statist. Prob. Lett. 1, 171–175.Google Scholar
Riordan, J. (1964). An Introduction to Combinatorial Analysis, 2nd edn. John Wiley, New York.Google Scholar
Schuster, E. F. (1991). Distribution theory of runs via exchangeable random variables. Statist. Prob. Lett. 11, 379–386.Google Scholar
Schwager, S. J. (1983). Run probabilities in sequences of Markov-dependence trials. J. Amer. Statist. Assoc. 78, 168–175.Google Scholar
Sen, K., Agarwal, M. and Bhattacharya, S. (2003). On circular distributions of order k based on Pólya– Eggenberger sampling scheme. J. Math. Sci. 2, 34–54.Google Scholar
Sen, K., Agarwal, M. and Chakraborty, S. (2002). Lengths of runs and waiting time distributions by using Pólya–Eggenberger sampling scheme. Studia Sci. Math. Hung. 39, 309–332.Google Scholar
Tripsiannis, G. A. and Philippou, A. N. (1997). A new multivariate inverse Pólya distribution of order k . Commun. Statist. Theory Meth. 26, 149–158.Google Scholar
Wolfowitz, J. (1943). On the theory of runs with some applications to quality control. Ann. Math. Statist. 14, 280–288.Google Scholar