Abstract
Waiting time distributions of simple and compound runs and patterns have been studied and applied in various areas of statistics and appliedprobability. Most current results are derived under the assumption that the sequence is either independent and identically distributed or first order Markov dependent. In this manuscript, we provide a comprehensive theoretical framework for obtaining waiting time distributions under the most general structure where the sequence consists of r-th order (r≥ 1) Markov dependent multi-state trials. We also show that the waiting time follows a generalized geometric distribution in terms of the essential transition probability matrix of the corresponding imbedded Markov chain. Further, we derive a large deviation approximation for the tail probability of the waiting time distribution based on the largest eigenvalue of the essential transition probability matrix. Numerical examples are given to illustrate the simplicity and efficiency of the theoretical results.
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References
Aki S., Hirano K. (1999). Sooner and later waiting time problems for runs in Markov dependent bivariate trials. Annals of the Institute of Statistical Mathematics 51:17–29
Aki S., Hirano K. (2004). Waiting time problems for a two-dimensional pattern. Annals of the Institute of Statistical Mathematics 56:169–182
Balakrishnan N., Koutras M.V. (2002). Runs and scans with applications. John Wiley & Sons, New York
Feller W. (1968). An introduction to probability theory and its applications (Vol. I, 3rd ed.). Wiley, New York
Fu J.C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials. Statistica Sinica 6:957–974
Fu J.C., Koutras M.V. (1994). Distribution theory of runs: A Markov chain approach. Journal of the American Statistical Association 89:1050–1058
Fu J.C., Chang Y.M. (2002). On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials. Journal of Applied Probability 39:70–80
Fu, J. C., Lou, W. Y. W. (2003). Distribution theory of runs and patterns and its applications: a finite Markov chain imbedding approach. World Scientific Publishing Co.
Han Q., Aki S. (2000a). Sooner and later waiting time problems based on a dependent sequence. Annals of the Institute of Statistical Mathematics 52:407–414
Han Q., Aki S. (2000b). Waiting time problems in a two-state Markov chain. Annals of the Institute of Statistical Mathematics 52:778–789
Han Q., Hirano K. (2003). Sooner and later waiting time problems for patterns in Markov dependent trials. Journal of Applied Probability 40:73–86
Koutras M.V. (1997). Waiting time distributions associated with runs of fixed length in two-state Markov chains. Annals of the Institute of Statistical Mathematics 49:123–139
Koutras M.V., Alexandrou V. (1995). Runs, scans and urn model distributions: a unified Markov chain approach. Annals of the Institute of Statistical Mathematics 47:743–766
Koutras M.V., Alexandrou V.A. (1997). Sooner waiting time problems in a sequence of trinary trials. Journal of Applied Probability 34:593–609
Lou W.Y.W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding. Journal of the American Statistical Association 91:1595–1601
Seneta E. (1981). Non-negative matrices and Markov chains, 2nd edition. Springer-Verlag, Berlin Heidelberg New York
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Fu, J.C., Lou, W.Y.W. Waiting Time Distributions of Simple and Compound Patterns in a Sequence of r-th Order Markov Dependent Multi-state Trials. Ann Inst Stat Math 58, 291–310 (2006). https://doi.org/10.1007/s10463-006-0038-8
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DOI: https://doi.org/10.1007/s10463-006-0038-8