Abstract
In this paper we study exact distributions of sooner and later waiting times for runs in Markov dependent bivariate trials. We give systems of linear equations with respect to conditional probability generating functions of the waiting times. By considering bivariate trials, we can treat very general and practical waiting time problems for runs of two events which are not necessarily mutually exclusive. Numerical examples are also given in order to illustrate the feasibility of our results.
Similar content being viewed by others
REFERENCES
Aki, S., Balakrishnan, N. and Mohanty, S. G. (1996). Sooner and later waiting time problems and failure runs in higher order Markov dependent trials, Ann. Inst. Statist. Math., 48, 773-787.
Aki, S. and Hirano, K. (1993). Discrete distributions related to succession events in a two-state Markov chain, Statistical Science & Data Analysis (eds. K. Matusita, M. L. Puri and T. Hayakawa), 467-474, VSP Publishers, Amsterdam.
Balakrishnan, N., Balasubramanian, K. and Viveros, R. (1995). Start-up demonstration tests under correlation and corrective action, Naval Res. Logist., 42, 1271-1276.
Balakrishnan, N., Mohanty, S. and Aki, S. (1997). Start-up demonstration tests under Markov dependence model with corrective actions, Ann. Inst. Statist. Math., 49, 155-169.
Balasubramanian, K., Viveros, R. and Balakrishnan, N. (1993). Sooner and later waiting time problems for Markovian Bernoulli trials, Statist. Probab. Lett., 18, 153-161.
Ebneshahrashoob, M. and Sobel, M. (1990). Sooner and later problems for Bernoulli trials: frequency and run quotas, Statist. Probab. Lett., 9, 5-11.
Hahn, G. J. and Gage, J. B. (1983). Evaluation of a start-up demonstration test, Journal of Quality Technology, 15, 103-106.
Ling, K. D. (1992). A generalization of the sooner and later waiting time problems for Bernoulli trials: frequency quota, Statist. Probab. Lett., 14, 401-405.
Sobel, M. and Ebneshahrashoob, M. (1992). Quota sampling for multinomial via Dirichlet, J. Statist. Plann. Inference, 33, 157-164.
Uchida, M. and Aki, S. (1995). Sooner and later waiting time problems in two-state Markov chain, Ann. Inst. Statist. Math., 47, 415-433.
Viveros, R. and Balakrishnan, N. (1993). Statistical inference from start-up demonstration test data, Journal of Quality Technology, 25, 119-130.
Yamamoto, H. and Miyakawa, M. (1995). Reliability of a linear connected-(r, s)-out-of-(m, n):F lattice system, IEEE Transactions on Reliability, 44, 333-336.
Author information
Authors and Affiliations
About this article
Cite this article
Aki, S., Hirano, K. Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials. Annals of the Institute of Statistical Mathematics 51, 17–29 (1999). https://doi.org/10.1023/A:1003874900507
Issue Date:
DOI: https://doi.org/10.1023/A:1003874900507