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On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials. (English) Zbl 1028.60068
Summary: Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. The exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

MSC:
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F10 Large deviations
60E05 Probability distributions: general theory
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[1] Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications . John Wiley, New York. · Zbl 0991.62087
[2] Barbour, A. D., Chryssaphinou, O. and Roos, M. (1995). Compound Poisson approximation in reliability theory. IEEE Trans. Reliab. 44, 398–402. · Zbl 04527633 · doi:10.1109/24.406572
[3] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximations . Oxford University Press. · Zbl 0746.60002
[4] Boutsikas, M. V. and Koutras, M. V. (2000). Reliability approximation for Markov chain imbeddable systems. Methodology Comput. Appl. Prob. 2, 393–411. · Zbl 0984.60091 · doi:10.1023/A:1010062218369
[5] Burr, E. J. and Cane, G. (1961). Longest run of consecutive observations having a special attribute. Biometrika 48, 461–465. · Zbl 0134.15002 · doi:10.1093/biomet/48.3-4.461
[6] Chao, M. T. and Fu, J. C. (1989). A limit theorem of certain repairable systems. Ann. Inst. Statist. Math. 41, 809–818. · Zbl 0692.60075 · doi:10.1007/BF00057742
[7] Chao, M. T. and Fu, J. C. (1991). The reliability of a large series system under the Markov structure. Adv. Appl. Prob. 23, 894–908. · Zbl 0795.60082 · doi:10.2307/1427682
[8] Chryssaphinou, O. and Papastavridis, S. G. (1990). Limit distribution for a consecutive-\(k\)-out-of-\(n\): F system. Adv. Appl. Prob. 22, 491–493. · Zbl 0713.60093 · doi:10.2307/1427550
[9] Derman, C., Liberman, G. J. and Ross, S. (1982). On the consecutive-\(k\)-out-of-\(n\):F systems. IEEE Trans. Reliab. 31, 57–63. · Zbl 0478.90029 · doi:10.1109/TR.1982.5221229
[10] Doi, M. and Yamamoto, E. (1998). On the joint distribution of runs in a sequence of multi-state trials. Statist. Prob. Lett. 39, 133–141. · Zbl 0910.60054 · doi:10.1016/S0167-7152(98)00051-0
[11] Erd\Hos, P. and Révész, P. (1975). On the length of the longest head run. In Topics in Information Theory (Colloq. Math. Soc. János Bolyai 16 ), eds I. Csiszár and P. Elias, North-Holland, Amsterdam, pp. 219–228.
[12] Fu, J. C. (1986). Bounds for reliability of large consecutive-\(k\)-out-of-\(n\): F systems with unequal component probabilities. IEEE Trans. Reliab. 35, 316–319. · Zbl 0597.90036 · doi:10.1109/TR.1986.4335442
[13] Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials. Statist. Sinica 6, 957–974. · Zbl 0857.60068
[14] Fu, J. C. and Chang, Y. M. (2002). On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials. J. Appl. Prob. 39, 70–80. · Zbl 1008.60031 · doi:10.1239/jap/1019737988
[15] Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Soc. 89, 1050–1058. · Zbl 0806.60011 · doi:10.2307/2290933
[16] Gibbons, J. D. (1971). Nonparametric Statistical Inference . McGraw-Hill, New York. · Zbl 0223.62050
[17] Godbole, A. P. and Schaffner, A. A. (1993). Improved Poisson approximations for word patterns. Adv. Appl. Prob. 25, 334–347. · Zbl 0772.60013 · doi:10.2307/1427656
[18] Goncharov, V. L. (1944). On the field of combinatory analysis. Izv. Akad. Nauk. SSSR Ser. Mat. 8 , 3–48 (in Russian). English translation: Amer. Math. Soc. Transl. 19 (1962), 1–46. · Zbl 0129.31503
[19] Gordon, L., Schilling, M. F. and Waterman, M. S. (1986). An extreme value theory for long head runs. Prob. Theory Relat. Fields 72, 279–287. · Zbl 0587.60031 · doi:10.1007/BF00699107
[20] Hirano, K. (1986). Some properties of the distributions of order \(k\). In Fibonacci Numbers and Their Applications (Patras, 1984), Reidel, Dordrecht, pp. 43–53. · Zbl 0601.62023
[21] Koutras, M. V. (1997). Waiting time distributions associated with runs of fixed length in two-state Markov chains. Ann. Inst. Statist. Math. 49, 123–139. · Zbl 0913.60019 · doi:10.1023/A:1003118807148
[22] Koutras, M. V. and Alexandrou, V. A. (1997). Sooner waiting time problems in a sequence of trinary trials. J. Appl. Prob. 34, 593–609. · Zbl 0891.60020 · doi:10.2307/3215087
[23] Lou, W. Y. W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding. J. Amer. Statist. Soc. 91, 1595–1601. · Zbl 0881.62086 · doi:10.2307/2291585
[24] Muselli, M. (2000). New improved bounds for reliability of consecutive-\(k\)-out-of-\(n\):F systems. J. Appl. Prob. 37, 1164–1170. · Zbl 0985.60080 · doi:10.1239/jap/1014843097
[25] Papastavridis, S. G. and Koutras, M. V. (1993). Bounds for reliability of consecutive-\(k\)-within-\(m\)-out-of-\(n\) systems. IEEE Trans. Reliab. 42, 156–160. · Zbl 0775.90194 · doi:10.1109/24.210288
[26] Petrov, V. V. (1965). On the probabilities of large deviations for sums of independent random variables. Theory Prob. Appl. 10, 287–298. · Zbl 0235.60028 · doi:10.1137/1110033
[27] Philippou, A. N. and Makri, F. S. (1986). Success runs and longest runs. Statist. Prob. Lett. 4, 211–215. · Zbl 0594.62013 · doi:10.1016/0167-7152(86)90069-6
[28] Rényi, A. (1970). Probability Theory . Academic Kiadó, Budapest. · Zbl 0233.60001
[29] Seneta, E. (1981). Nonnegative Matrices and Markov chains , 2nd edn. Springer, New York. · Zbl 0471.60001
[30] Schilling, M. F. (1990). The longest run of heads. College Math. J. 21, 196–207. · Zbl 0995.60502 · doi:10.2307/2686886
[31] Suman, K. A. (1994). The longest run of any letter in a randomly generated word. In Runs and Patterns in Probability: Selected Papers (Math. Appl. 283 ), eds A. P. Godbole and S. G. Papastavridis, Kluwer, Dordrecht, pp. 119–130. · Zbl 0832.60014
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