Abstract
In this article, waiting time distributions of compound patterns are considered in terms of the generating function of the numbers of occurrences of the compound patterns. Formulae for the evaluation of the generating functions of waiting time are given, which are very effective computational tools. We provide several viewpoints on waiting time problems associated with compound patterns and develop a general workable framework for the study of the corresponding distributions. The general theory is employed for the investigation of some examples in order to illustrate how the distributions of waiting time can be derived through our theoretical results.
Similar content being viewed by others
References
Antzoulakos D.L. (2001). Waiting times for patterns in a sequence of multistate trials. Journal of Applied Probability 38, 508–518
Balakrishnan N., Koutras M.V. (2002). Runs and scans with applications. New York, Wiley
Chao M.T., Fu J.C., Koutras M.V. (1995). Survey of reliability studies of consecutive-k-out-of-n: F & related systems. IEEE Transactions on Reliability 40, 120–127
Chryssaphinou O., Papastavridis S. (1990). The occurrence of a sequence of patterns in repeated dependent experiments. Theory of Probability and its Applications 35, 167–173
Ewens W.J., Grant G.R. (2001). Statistical methods in bioinformatics: an introduction. Heidelberg, Springer.
Feller W. (1968). An introduction to probability theory and its applications, Vol. I (3rd ed). New York, Wiley
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., 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., Koutras M.V. (1994). Distribution theory of runs: a Markov chain approach. Journal of the American Statistical Association 89, 1050–1058
Fu J.C., Lou W.Y.W. (2003). Distribution theory of runs and patterns and its applications: a finite markov chain imbedding approach. Singapore, World Scientific
Glaz J., Naus J., Wallenstein S. (2001). Scan statistics. New York, Springer
Graham R.L., Knuth D.E., Patashnik O. (1994). Concrete mathematics (2nd ed). Reading, Addison-Wesley
Han Q., Hirano K. (2003). Sooner and later waiting time problems for patterns in Markov dependent trials. Journal of Applied Probability 40, 73–86
Hirano K., Aki S. (2003). Number of occurrences of subpattern until the first appearance of a pattern and geometric distribution. Statistics & Probability Letters 65, 259–262
Inoue, K. (2004). Joint distributions associated with patterns, successes and failures in a sequence of multi-state trials. Annals of the Institute of Statistical Mathematics 56, 143–168
Inoue K., Aki S. (2002). Generalized waiting time problems associated with pattern in Polya’s urn scheme. Annals of the Institute of Statistical Mathematics 54, 681–688
Inoue K., Aki S. (2005a). A generalized Pólya urn model and related multivariate distributions. Annals of the Institute of Statistical Mathematics 57, 49–59
Inoue, K., Aki, S. (2005b). On generating functions of waiting times and numbers of occurrences of compound patterns in a sequence of multi-state trials. Research Memorandum, No. 949, The Institute of Statistical Mathematics, Japan.
Johnson N.L., Kotz S. (1977). Urn models and their applications. New York, Wiley
Koutras M.V. (1997). Waiting times and number of appearances of events in a sequence of discrete random variables. In: Balakrishnan N. (eds), Advances in combinatorial methods and applications to probability and statistics. Boston, Birkhauser, pp. 363–384
Krauth J. (1999). Ratchet scan and disjoint scan statistics. In: Glaz J., Balakrishnan N. (eds), Scan statistics and applications. Boston, Birkhauser, pp. 67–96
Robin S., Daudin J.J. (1999). Exact distribution of word occurrences in a random sequences of letters. Journal of Applied Probability 36, 179–193
Robin S., Daudin J.J. (2001). Exact distribution of the distances between any occurrences of a set of words. Annals of the Institute of Statistical Mathematics 53, 895–905
Shmueli G., Cohen A. (2000). Run-related probability functions applied to sampling inspection. Technometrics 42, 188–202
Stefanov V.T. (2000). On some waiting time problems. Journal of Applied Probability 37, 756–764
Stefanov V.T. (2003). The intersite distances between pattern occurrences in strings generated by general discrete—and continuous—time models: an algorithmic approach. Journal of Applied Probability 40, 881–892
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by the ISM Cooperative Research Program (2006-ISM·CRP-2007).
About this article
Cite this article
Inoue, K., Aki, S. On waiting time distributions associated with compound patterns in a sequence of multi-state trials. Ann Inst Stat Math 61, 499–516 (2009). https://doi.org/10.1007/s10463-007-0140-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-007-0140-6