Recursive equations in finite Markov chain imbedding
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
In this paper, recursive equations for waiting time distributions of r-th occurrence of a compound pattern are studied via the finite Markov chain imbedding technique under overlapping and non-overlapping counting schemes in sequences of independent and identically distributed (i.i.d.) or Markov dependent multi-state trials. Using the relationship between number of patterns and r-th waiting time, distributions of number of patterns can also be obtained. The probability generating functions are also obtained. Examples and numerical results are given to illustrate our theoretical results.
- Aki, S. (1992). Waiting time problems for a sequence of discrete random variables. Annals of the Institute of Statistical Mathematics, 44, 363–378. CrossRef
- Chang, Y. M. (2005). Distribution of waiting time until the \(r\) th occurrence of acompound pattern. Statistics and Probability Letters, 75, 29–38. CrossRef
- Chang, Y. M., Wu, T. L. (2011). On average run lengths of control charts for autocorrelated processes. Methodology and Computing in Applied Probability, 13, 419–431.
- Chang, Y. M., Fu, J. C., Lin, H. Y. (2012). Distribution and double generating function of number of patterns in a sequence of Markov dependent multistate trials. Annals of the Institute of Statistical Mathematics, 64, 55–68.
- Cui, L., Xu, Y., Zhao, X. (2010). Developments and applications of the finite Markov chain imbedding approach in reliability. IEEE Transactions on Reliability, 59, 685–690.
- 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., Lou, W. Y. W. (2003). Distribution theory of runs and patterns and its applications. River Edge, NJ: World Scientific Publishing Co. Inc.
- Fu, J. C., Lou, W. Y. W., Wang, Y. J. (1999). On the exact distributions of Eulerian and Simon Newcomb numbers associated with random permutations. Statistics and Probability Letters, 42, 115–125.
- Fu, J. C., Shmueli, G., Chang, Y. M. (2003). A unified Markov chain approach for computing the run length distribution in control charts with simple or compound rules. Statistics and Probability Letters, 65, 457–466.
- 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. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain. Statistica Sinica, 3, 313–320.
- Inoue, K., Aki, S. (2005). A generalized Pólya urn model and related multivariate distributions. Annals of the Institute of Statistical Mathematics, 57, 49–59.
- Inoue, K., Aki, S. (2007). On generating functions of waiting times and numbers of occurrences of compound patterns in a sequence of multistate trials. Journal of Applied Probability, 44, 71–81.
- Inoue, K., Aki, S. (2009). On waiting time distributions associated with compound patterns in a sequence of multi-state trials. Annals of the Institute of Statistical Mathematics, 61, 499–516.
- Koutras, M. V. (1997). Waiting times and number of appearances of events in a sequence of discrete random variables (pp. 363–384). In Advances in combinatorial methods and applications to probability and statistics. Statistics for Industry and Technology. Boston, MA: Birkhäuser Boston
- Koutras, M. V., Milienos, F. S. (2012). Exact and asymptotic results for pattern waiting times. Journal of Statistical Planning and Inference, 142, 1464–1479.
- 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. CrossRef
- Nuel, G. (2008). Pattern Markov chains: Optimal Markov chain embedding through deterministic finite automata. Journal of Applied Probability, 45, 226–243. CrossRef
- Zhao, X., Cui, L. (2009). On the accelerated scan finite Markov chain imbedding approach. IEEE Transactions on Reliability, 58, 383–388.
- Recursive equations in finite Markov chain imbedding
Annals of the Institute of Statistical Mathematics
Volume 65, Issue 3 , pp 513-527
- Cover Date
- Print ISSN
- Online ISSN
- Springer Japan
- Additional Links
- Recursive equation
- Simple and compound patterns
- Waiting time
- Finite Markov chain imbedding
- Probability generating function
- Industry Sectors