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Run Statistics Defined on the Multicolor URN Model

Published online by Cambridge University Press:  14 July 2016

Serkan Eryilmaz*
Affiliation:
Izmir University of Economics
*
Postal address: Izmir University of Economics, Department of Mathematics, 35330 Izmir, Turkey. Email address: [email protected]
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Abstract

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Recently, Makri, Philippou and Psillakis (2007b) studied the exact distribution of success run statistics defined on an urn model. They derived the exact distributions of various success run statistics for a sequence of binary trials generated by the Pólya-Eggenberger sampling scheme. In our study we derive the joint distributions of run statistics defined on the multicolor urn model using a simple unified combinatorial approach and extend some of the results of Makri, Philippou and Psillakis (2007b). As a consequence of our results, we obtain the joint distributions of success and failure runs defined on the two-color urn model. The results enable us to compute the characteristics of particular consecutive-type systems and start-up demonstration tests.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2008 

References

Antzoulakos, D. L. and Boutsikas, M. V. (2007). A direct method to obtain the Joint distribution of successes, failures and patterns in enumeration problems. Statist. Prob. Lett. 77, 3239.Google Scholar
Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Chakraborti, S. and Eryılmaz, S. (2007). A nonparametric Shewhart-type signed-rank control chart based on runs. Commun. Statist. Simul. Comput. 36, 335356.CrossRefGoogle Scholar
Charalambides, C. A. (2002). Enumerative Combinatorics. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
Chang, C. J., Fann, C. S. J., Chou, W. C. and Lian, I. B. (2003). On the tail probability of the longest well-matching run. Statist. Prob. Lett. 63, 267274.Google Scholar
Chao, M. T., Fu, J. C. and Koutras, M. V. (1995). Survey of reliability studies of consecutive k-out-of-n: F and related systems. IEEE Trans. Reliab. 44, 120127.CrossRefGoogle Scholar
Eryılmaz, S. (2008a). Consecutive k-out-of-n:G system in stress-strength setup. Commun. Statist. Simul. Comput. 37, 579589.Google Scholar
Eryılmaz, S. (2008b). Distribution of runs in a sequence of exchangeable multistate trials. Statist. Prob. Lett. 78, 15051513.Google Scholar
Eryılmaz, S. and Demir, S. (2007). Success runs in a sequence of exchangeable binary trials. J. Statist. Planning Infer. 137, 29542963.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. (1996). Distribution theory of runs and patterns associated with a sequence of multistate trials. Statistica Sinica 6, 957974.Google Scholar
Fu, J. C. and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications. World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Fu, J. C. and Lou, W. Y. W. (2007). On the normal approximation for the distribution of the number of simple or compound patterns in a random sequence of multistate trials. Methodology Comput. Appl. Prob. 9, 195205.CrossRefGoogle Scholar
Fu, J. C., Shmueli, G. and Chang, Y. M. (2003). A unified Markov chain approach for computing the run length distribution in control charts with simple or compound rules. Statist. Prob. Lett. 65, 457466.Google Scholar
Goldstein, L. (1990). Poisson approximations and DNA sequence matching. Commun. Statist. Theory Meth. 19, 41674179.Google Scholar
Han, Q. and Aki, S. (1999). Joint distributions of runs in a sequence of multistate trials. Ann. Inst. Statist. Math. 51, 419447.CrossRefGoogle Scholar
Hill, B. M., Lane, D. and Sudderth, W. (1987). Exchangeable urn processes. Ann. Prob. 15, 15861592.CrossRefGoogle Scholar
Inoue, K. and Aki, S. (2005). Joint distributions of numbers of success runs of specified lengths in linear and circular sequences. Ann. Inst. Statist. Math. 57, 353368.Google Scholar
Johnson, N. and Kotz, S. (1977). Urn Models and Their Application. John Wiley, New York.Google Scholar
Koutras, M. V. (1997). Consecutive-k,r-out-of-n: DFM systems. Microelectron. Reliab. 37, 597603.Google Scholar
Koutras, M. V. and Alexandrou, V. A. (1997). Non-parametric tests based on success runs of fixed length. Statist. Prob. Lett. 32, 393404.Google Scholar
Kong, Y. (2006). Distribution of runs and longest runs: a new generating function approach. J. Amer. Statist. Assoc. 101, 12531263.Google Scholar
Lau, T. S. (1992). The reliability of exchangeable binary systems. Statist. Prob. Lett. 13, 153158.CrossRefGoogle Scholar
Lou, W. Y. W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding. J. Amer. Statist. Assoc. 91, 15951601.Google Scholar
Lou, W. Y. W. (2003). The exact distribution of the k-tuple statistic for sequence homology. Statist. Prob. Lett. 61, 5159.Google Scholar
Makri, F. S. and Philippou, A. N. (2005). On binomial and circular binomial distributions of order k for l-overlapping success runs of length k. Statist. Papers 46, 411432.Google Scholar
Makri, F. S., Philippou, A. N. and Psillakis, Z. M. (2007a). Shortest and longest length of success runs in binary sequences. J. Statist. Planning Infer. 137, 22262239.Google Scholar
Makri, F. S., Philippou, A. N. and Psillakis, Z. M. (2007b). Success run statistics defined on an urn model. Adv. Appl. Prob. 39, 9911019.Google Scholar
Mosteller, F. (1941). Note on an application of runs to quality control charts. Ann. Math. Statist. 12, 228232.Google Scholar
Philippou, A. N. (1988). Recursive theorems for success runs and reliability of consecutive-k-out-of-n: F systems. In Applications of Fibonacci Numbers, Kluwer, Dordrecht, pp. 149161.Google Scholar
Philippou, A. N. and Makri, F. S. (1985). Longest success runs and Fibonacci-type polynomials. Fibonacci Quart. 23, 338346.Google Scholar
Philippou, A. N. and Makri, F. S. (1986). Successes, runs and longest runs. Statist. Prob. Lett. 4, 101105.Google Scholar
Singpurwalla, N. D. (2006). Reliability and Risk. John Wiley, Chichester.Google Scholar
Sen, K., Agarwal, M. L. and Chakraborty, S. (2002). Lengths of runs and waiting time distributions by using Pólya–Eggenberger sampling scheme. Studia Sci. Math. Hung. 39, 309332.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, 3454.Google Scholar
Smith, M. L. and Griffith, S. W. (2008). The analysis and comparison of start-up demonstration tests. Europ. J. Operat. Res. 186, 10291045.Google Scholar
Vaggelatou, E. (2003). On the length of the longest run in a multi-state Markov chain. Statist. Prob. Lett. 62, 211221.Google Scholar
Wald, A. and Wolfowitz, J. (1940). On a test whether two samples are from the same population. Ann. Math. Statist. 11, 147162.Google Scholar
Wang, L. and Cheung, K. (2004). Use of run statistics for pattern recognition in genomic DNA sequences. J. Comput. Biol. 11, 107124.Google Scholar
Wolfowitz, J. (1943). On the theory of runs with some applications to quality control. Ann. Math. Statist. 14, 280288.CrossRefGoogle Scholar
Zabell, S. L. (1982). Johnson‘s “sufficientness” postulate. Ann. Statist. 10, 10911099.CrossRefGoogle Scholar