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Joint distributions of successes, failures and patterns in enumeration problems

Part of: Distribution theory - Probability Special processes Distribution theory Markov processes

Published online by Cambridge University Press:  01 July 2016

S Chadjiconstantinidis ,
D. L. Antzoulakos  and
M. V. Koutras
S Chadjiconstantinidis*
Affiliation:
University of Piraeus
D. L. Antzoulakos*
Affiliation:
University of Piraeus
M. V. Koutras*
Affiliation:
University of Athens
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Piraeus, Piraeus 18534, Greece.
∗ Postal address: Department of Statistics and Actuarial Science, University of Piraeus, Piraeus 18534, Greece.
∗ Postal address: Department of Statistics and Actuarial Science, University of Piraeus, Piraeus 18534, Greece.
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Abstract

Let ε be a (single or composite) pattern defined over a sequence of Bernoulli trials. This article presents a unified approach for the study of the joint distribution of the number Sn of successes (and Fn of failures) and the number Xn of occurrences of ε in a fixed number of trials as well as the joint distribution of the waiting time Tr till the rth occurrence of the pattern and the number STr of successes (and FTr of failures) observed at that time. General formulae are developed for the joint probability mass functions and generating functions of (Xn,Sn), (Tr,STr) (and (Xn,Sn,Fn),(Tr,STr,FTr)) when Xn belongs to the family of Markov chain imbeddable variables of binomial type. Specializing to certain success runs, scans and pattern problems several well-known results are delivered as special cases of the general theory along with some new results that have not appeared in the statistical literature before.

Keywords

Markov chainswaiting timessooner waiting time problemspatternsscanssuccess runs

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K30: Applications (congestion, allocation, storage, traffic, etc.) 62E15: Exact distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 866 - 884
Copyright
Copyright © Applied Probability Trust 2000 

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References

Aki, S. (1985). Discrete distributions of order k on a binary sequence. Ann. Inst. Statist. Math. 37, 205–224.CrossRefGoogle Scholar
Aki, S. and Hirano, K. (1994). Distributions of numbers of failures and successes until the first consecutive k successes. Ann. Inst. Statist. Math. 46, 193–202.CrossRefGoogle Scholar
Aki, S. and Hirano, K. (1995). Joint distributions of numbers of success-runs and failures until the first consecutive k successes. Ann. Inst. Statist. Math. 47, 225–235.CrossRefGoogle Scholar
Aki, S., Balakrishnan, N. and Mohanty, S. G. (1996). Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials. Ann. Inst. Statist. Math. 48, 773–787.CrossRefGoogle Scholar
Antzoulakos, D. L. (1999). On waiting time problems associated with runs in Markov dependent trials. Ann. Inst. Statist. Math. 51, 323–330.CrossRefGoogle Scholar
Antzoulakos, D. L. and Chadjiconstantinidis, S. (2000). Distributions of numbers of success runs of fixed length in Markov dependent trials. To appear in Ann. Inst. Statist. Math.Google Scholar
Antzoulakos, D. L. and Philippou, A. N. (1996a). Distributions of the numbers of successes and failures until the occurrence of the rth overlapping and non-overlapping success run of length k. J. Indian Soc. Statist. Operat. Res. 17, 19–30.Google Scholar
Antzoulakos, D. L. and Philippou, A. N. (1996b). Derivation of the probability distribution functions for succession quota random variables. Ann. Inst. Statist. Math. 48, 551–561.CrossRefGoogle Scholar
Balakrishnan, N. (1997). Joint distributions of numbers of success-runs and failures until the first consecutive k successes in a binary sequence. Ann. Inst. Statist. Math. 49, 519–529.CrossRefGoogle Scholar
Balakrishnan, N. and Koutras, M. V. (2000). Runs, Scans and Applications. John Wiley, New York (in preparation).Google Scholar
Chadjiconstantinidis, S. and Koutras, M. V. (2000). Distributions of the numbers of failures and successes in a waiting time problem. To appear in Ann. Inst. Statist. Math.Google Scholar
Chen, J. and Glaz, J. (1997). Approximations and inequalities for the distribution of a scan statistic for 0-1 Bernoulli trials, In Advances in the Theory and Practice of Statistics, eds Johnson, N. L. and Balakrishnan, N.. John Wiley, New York, pp. 285–298.Google Scholar
Chen, J. and Glaz, J. (1999). Approximations for the distribution and the moments of discrete scan statistics. In Scan Statistics and Applications, eds Glaz, J. and Balakrishnan, N.. Birkhäuser, Boston, pp. 27–66.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Ebneshahrashoob, M. and Sobel, M. (1990). Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas. Statist. Prob. Lett. 9, 5–11.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multistate trials. Statist. Sinica 6, 957–974.Google Scholar
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058.CrossRefGoogle Scholar
Glaz, J. and Balakrishnan, N. (1999). Introduction to scan statistics. In Scan Statistics and Applications, eds Glaz, J. and Balakrishnan, N.. Birkhäuser, Boston, pp. 3–24.CrossRefGoogle Scholar
Han, Q. and Aki, S. (1998). Formulae and recursion for the joint distributions of success runs of several lengths in a two-state Markov chain. Statist. Prob. Lett. 40, 203–214.CrossRefGoogle Scholar
Han, Q. and Aki, S. (1999). Joint distributions of runs in a sequence of multistate trials. Ann. Inst. Statist. Math. 51, 419–447.CrossRefGoogle Scholar
Hirano, K., Aki, S. and Uchida, M. (1997). Distributions of numbers of success-runs until the first consecutive k successes in higher order Markov dependent trials. In Advances in Combinatorial Methods and Applications to Probability and Statistics, ed. Balakrishnan, N.. Birkhäuser, Boston, pp. 401–410.CrossRefGoogle Scholar
Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions, 2nd edn. John Wiley, New York.Google Scholar
Koutras, M. V. (1996). On a waiting time distribution in a sequence of Bernoulli trials. Ann. Inst. Statist. Math. 48, 789–806.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Koutras, M. V. (2000). Applications of Markov chains to the distribution theory of runs and patterns. In Handbook of Statistics, Vol. 20. Stochastic Processes: Modelling and Simulation, eds Rao, C. R. and Shanbhag, D. N.. North Holland, Amsterdam, pp.Google Scholar
Koutras, M. V. and Alexandrou, V. A. (1995). Runs, scans and urn model distributions: a unified Markov chain approach. Ann. Inst. Statist. Math. 47, 743–766.CrossRefGoogle Scholar
Koutras, M. V. and Alexandrou, V. A. (1997). Sooner waiting time problems in a sequence of trinary trials. J. Appl. Prob. 34, 593–609.CrossRefGoogle Scholar
Ling, K. D. (1989). A new class of negative binomial distributions of order k. Statist. Prob. Lett. 7, 371–376.CrossRefGoogle Scholar
Mood, A. M. (1940). The distribution theory of runs. Ann. Math. Statist. 11, 367–392.CrossRefGoogle Scholar
Philippou, A. N. (1984). The negative binomial distribution of order k and some of its properties. Biometrical J. 26, 789–794.CrossRefGoogle Scholar
Uchida, M. (1998). Joint distributions of the numbers of success runs until the first consecutive k successes in a higher-order two-state Markov chain. Ann. Inst. Statist. Math. 50, 203–222.CrossRefGoogle Scholar