Distributions of runs and scans in multistate Markov exchangeable sequences

References

• Aki, S. 2012. Statistical modeling for discrete patterns in a sequence of exchangeable trials. Annals of the Institute of Statistical Mathematics 64 (3):633–55. doi:10.1007/s10463-011-0325-x.
   Web of Science ®Google Scholar
• Aki, S., and K. Hirano. 2000. Numbers of success-runs of specified length until certain stopping time rules and generalized binomial distributions of order k. Annals of the Institute of Statistical Mathematics 52 (4):767–77. doi:10.1023/A:1017585512412.
   Web of Science ®Google Scholar
• Aki, S., and K. Hirano. 2013. Coupon collector’s problems with statistical applications to rankings. Annals of the Institute of Statistical Mathematics 65 (3):571–87. doi:10.1007/s10463-012-0382-9.
   Web of Science ®Google Scholar
• Antzoulakos, D. L. 2003. Waiting times and number of appearances of runs: A unified approach. Communications in Statistics - Theory and Methods 32 (7):1289–315. doi:10.1081/STA-120021560.
   Web of Science ®Google Scholar
• Antzoulakos, D. L., M. V. Koutras, and A. C. Rakitzis. 2009. Start-up demonstration tests based on run and scan statistics. Journal of Quality Technology 41 (1):48–59. doi:10.1080/00224065.2009.11917759.
   Web of Science ®Google Scholar
• Antzoulakos, D. L., and A. C. Rakitzis. 2008. The modified r out of m control chart. Communications in Statistics - Simulation and Computation 37 (2):396–408. doi:10.1080/03610910701501906.
   Web of Science ®Google Scholar
• Balakrishnan, N., and M. V. Koutras. 2002. Runs and scans with applications. New York: John Wiley.
   Google Scholar
• Balakrishnan, N., M. V. Koutras, and F. Milienos. 2021. Reliability analysis and plans for successive testing: Start-up demonstration tests and applications. London: Academic Press.
   Google Scholar
• Boutsikas, M. V., D. L. Antzoulakos, and A. C. Rakitzis. 2014. On the joint distribution of stopping times and stopped sums in multistate exchangeable trials. Journal of Applied Probability 51 (2):483–91. doi:10.1239/jap/1402578638.
   Web of Science ®Google Scholar
• Boutsikas, M. V., and M. V. Koutras. 2002. On a class of multiple failure mode systems. Naval Research Logistics 49 (2):167–85. doi:10.1002/nav.10007.
   Web of Science ®Google Scholar
• Chao, M. T., J. C. Fu, and M. V. Koutras. 1995. Survey of reliability studies of consecutive-k-out-of-n: F & related systems. IEEE Transactions on Reliability 40:120–7.
   Web of Science ®Google Scholar
• Cui, L., W. Kuo, J. Li, and M. Xie. 2006. On the dual reliability systems of (n, f, k) and 〈n,f,k〉. Statistics and Probability Letters 76:1081–8.
   Web of Science ®Google Scholar
• Dafnis, S. D., and F. S. Makri. 2022. Weak runs in sequences of binary trials. Metrika 85 (5):573–603. doi:10.1007/s00184-021-00842-1.
   Web of Science ®Google Scholar
• Diaconis, P., and D. Freedman. 1980. De Finetti’s theorem for Markov chain. Annals of Probability 8:115–30.
   Web of Science ®Google Scholar
• Eryilmaz, S., and S. Chakraborti. 2008. On start-up demonstration tests under exchangeability. IEEE Transactions on Reliability 57 (4):627–32. doi:10.1109/TR.2008.2006290.
   Web of Science ®Google Scholar
• Eryilmaz, S., and S. Demir. 2007. Success runs in a sequence of exchangeable binary trials. Journal of Statistical Planning and Inference 137 (9):2954–63. doi:10.1016/j.jspi.2006.10.015.
   Web of Science ®Google Scholar
• Eryilmaz, S., and F. Yalcin. 2011. Distribution of run statistics in partially exchangeable processes. Metrika 73:293–304.
   Web of Science ®Google Scholar
• Fortini, S., L. Ladelli, G. Petris, and E. Regazzini. 2002. On mixtures of distributions of Markov chains. Stochastic Processes and Their Applications 100 (1–2):147–65. doi:10.1016/S0304-4149(02)00093-5.
   Google Scholar
• Freedman, D. A. 1962. Invariants under mixing which generalize de Finetti’s theorem. The Annals of Mathematical Statistics 33 (3):916–23. doi:10.1214/aoms/1177704460.
   Google Scholar
• Fu, J. C. 1996. Distribution theory of runs and patterns associated with a sequence of multi-state trials. Statistica Sinica 6:957–74.
   Web of Science ®Google Scholar
• Fu, J. C., and M. V. Koutras. 1994. Distribution theory of runs: A Markov chain approach. Journal of the American Statistical Association 89 (427):1050–8. doi:10.1080/01621459.1994.10476841.
   Web of Science ®Google Scholar
• Fu, J. C., and W. Y. W. Lou. 2003. Distribution theory of runs and patterns and its applications: A finite Markov chain imbedding approach. Singapore: World Scientific.
   Google Scholar
• Glaz, J., J. Naus, and S. Wallenstein. 2001. Scan statistics. New York: Springer-Verlag.
   Google Scholar
• Inoue, K., and S. Aki. 2003. Generalized binomial and negative binomial distributions of order k by the ℓ-overlapping enumeration scheme. Annals of the Institute of Statistical Mathematics 55:153–67.
   Web of Science ®Google Scholar
• Inoue, K., and S. Aki. 2007. Joint distributions of numbers of runs of specified lengths in a sequence of Markov dependent multistate trials. Annals of the Institute of Statistical Mathematics 59 (3):577–95. doi:10.1007/s10463-006-0056-6.
   Web of Science ®Google Scholar
• Inoue, K., S. Aki, and K. Hirano. 2011. Distributions of simple patterns in some kinds of exchangeable sequence. Journal of Statistical Planning and Inference 141 (8):2532–44. doi:10.1016/j.jspi.2011.02.002.
   Web of Science ®Google Scholar
• Kong, Y. 2019. Decoupling combinatorial complexity: A two-step approach to distributions of runs. Methodology and Computing in Applied Probability 21 (3):789–803. doi:10.1007/s11009-018-9689-1.
   Web of Science ®Google Scholar
• Koutras, M. V., and V. A. Alexandrou. 1997. Sooner waiting time problems in a sequence of trinary trials. Journal of Applied Probability 34 (3):593–609. doi:10.2307/3215087.
   Web of Science ®Google Scholar
• Krauth, J. 1999. Ratchet scan and disjoint scan statistics. In Scan statistics and applications, ed. J. Glaz and N. Balakrishnan, 67–96. Boston, MA: Birkhäuser.
   Google Scholar
• Lau, T. S. 1992. The reliability of exchangeable binary systems. Statistics & Probability Letters 13 (2):153–8. doi:10.1016/0167-7152(92)90090-R.
   Web of Science ®Google Scholar
• Mahmoud, H. 2008. Pólya urn models. London: Chapman & Hall.
   Google Scholar
• Makri, F. S., A. N. Philippou, and Z. M. Psillakis. 2007. Success run statistics defined on an urn model. Advances in Applied Probability 39 (4):991–1019. doi:10.1239/aap/1198177236.
   Web of Science ®Google Scholar
• Quintana, F. A., and M. A. Newton. 1998. Assesting the order of dependence for partially exchangeable binary data. Journal of the American Statistical Association 93 (441):194–202. doi:10.1080/01621459.1998.10474101.
   Web of Science ®Google Scholar
• Quintana, F. A., and M. A. Newton. 1999. Parametric partially exchangeable models for modeling multiple binary sequences. Brazilian Journal of Probability and Statistics 13:55–76.
   Google Scholar
• Rakitzis, A. C., and D. L. Antzoulakos. 2015. Start-up demonstration tests with three-level classification. Statistical Papers 56 (1):1–21. doi:10.1007/s00362-013-0564-4.
   Web of Science ®Google Scholar
• Shmueli, G., and A. Cohen. 2000. Run-related probability functions applied to sampling inspection. Technometrics 42 (2):188–202. doi:10.1080/00401706.2000.10485998.
   Web of Science ®Google Scholar
• Zabell, S. L. 1995. Characterizing Markov exchangeable sequences. Journal of Theoretical Probability 8 (1):175–8. doi:10.1007/BF02213460.
   Web of Science ®Google Scholar