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On the Joint Distribution of Stopping Times and Stopped Sums in Multistate Exchangeable Trials

Part of: Stochastic processes Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  19 February 2016

M. V. Boutsikas ,
D. L. Antzoulakos  and
A. C. Rakitzis
M. V. Boutsikas*
Affiliation:
University of Piraeus
D. L. Antzoulakos*
Affiliation:
University of Piraeus
A. C. Rakitzis*
Affiliation:
University of Cyprus
*
∗ Postal address: Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece.
∗ Postal address: Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece.
∗∗∗∗ Current address: LUNAM Université, Université de Nantes, IRCCyN UMR CNRS 6597, France. Email address: [email protected].
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Abstract

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Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let ST,i be the stopped sum denoting the number of appearances of outcome 'i' in X1, …, XT, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, ST,0, ST,1, …, ST,m). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

Keywords

Stopping timestopped sumexchangeabilitymultistate trialrun and scan statisticsacceptance samplingcoupon collector's problem

MSC classification

Primary: 60E05: Distributions 62E15: Exact distribution theory
Secondary: 60G09: Exchangeability 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 483 - 491
Copyright
© Applied Probability Trust 

References

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.Google Scholar
Antzoulakos, D. L. and Boutsikas, M. V. (2007). A direct method to obtain the Joint distribution of successes, failures and patterns in enumeration problems. Statist. Prob. Lett. 77, 32–39.Google 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.Google Scholar
Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Boutsikas, M. V., Rakitzis, A. C. and Antzoulakos, D. L. (2011). On the relation between the distributions of stopping time and stopped sum with applications. Preprint. Available at http://arxiv.org/abs/1008.0116v2.Google Scholar
Chadjiconstantinidis, S., Antzoulakos, D. L. and Koutras, M. V. (2000). Joint distributions of successes, failures and patterns in enumeration problems. Adv. Appl. Prob. 32, 866–884.Google Scholar
Eryilmaz, S. (2008a). Distribution of runs in a sequence of exchangeable multi-state trials. Statist. Prob. Lett. 78, 1505–1513.Google Scholar
Eryilmaz, S. (2008b). Run statistics defined on the multicolor urn model. J. Appl. Prob. 45, 1007–1023.Google Scholar
Eryilmaz, S. (2010). Discrete scan statistics generated by exchangeable binary trials. J. Appl. Prob. 47, 1084–1092.Google Scholar
Inoue, K. (2004). Joint distributions associated with patterns, successes and failures in a sequence of multi-state trials. Ann. Inst. Statist. Math. 56, 143–168.Google Scholar
Inoue, K. and Aki, S. (2007). On generating functions of waiting times and numbers of occurrences of compound patterns in a sequence of multistate trials. J. Appl. Prob. 44, 71–81.CrossRefGoogle Scholar
Inoue, K., Aki, S. and Hirano, K. (2011). Distributions of simple patterns in some kinds of exchangeable sequences. J. Statist. Planning Infer. 141, 2532–2544.Google Scholar
Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Lou, W. Y. W. and Fu, J. C. (2009). On probabilities for complex switching rules in sampling inspection. In Scan Statistics, eds Glaz, J., Pozdnyakov, V. and Wallenstein, S., Birkhäuser, Boston, MA, pp. 203–219.Google Scholar
Makri, F. S. and Psillakis, Z. M. (2011). On success runs of length exceeded a threshold. Methodology Comput. Appl. Prob. 13, 269–305.CrossRefGoogle Scholar
Mauldin, R. D., Sudderth, W. D. and Williams, S. C. (1992). Pólya trees and random distributions. Ann. Statist. 20, 1203–1221.Google Scholar
Uchida, M. (1998). On generating functions of waiting time problems for sequence patterns of discrete random variables. Ann. Inst. Statist. Math. 50, 655–671.Google Scholar