Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-09T07:12:21.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete Scan Statistics Generated by Exchangeable Binary Trials

Part of: Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  14 July 2016

Serkan Eryilmaz
Serkan Eryilmaz*
Affiliation:
Izmir University of Economics
*
∗Current address: Department of Industrial Engineering, Atilim University, 06836 Incek, Ankara, Turkey. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {Xi}i=1n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable Sn,m to be the maximum number of 1s within any m consecutive trials in {Xi}i=1n. The random variable Sn,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {Xi}i=1n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of Sn,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

Keywords

Bernoulli sequenceexchangeable binary trialreliabilityscan statistics

MSC classification

Secondary: 60E05: Distributions 62E15: Exact distribution theory
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1084 - 1092
Copyright
Copyright © Applied Probability Trust 2010 

References

Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603.CrossRefGoogle Scholar
Eryılmaz, S. (2008a). Distribution of runs in a sequence of exchangeable multi-state trials. Statist. Prob. Lett. 78, 1505–1513.CrossRefGoogle Scholar
Eryılmaz, S. (2008b). Run statistics defined on the multicolor urn model. J. Appl. Prob. 45, 1007–1023.Google Scholar
Eryılmaz, S. (2010). Mixture representations for the reliability of consecutive-k systems. Math. Comput. Modelling 51, 405–412.Google Scholar
Eryılmaz, S. and Demir, S. (2007). Success runs in a sequence of exchangeable binary trials. J. Statist. Planning Infer. 137, 2954–2963.Google Scholar
George, E. O. and Bowman, D. (1995). A full likelihood procedure for analysing exchangeable binary data, Biometrics 51, 512–523.Google Scholar
Glaz, J., Naus, J. and Wallanstein, S. (2001). Scan Statistics, Springer, New York.Google Scholar
Haiman, G. (2007). Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences. J. Statist. Planning Infer. 137, 821–828.Google Scholar
Inoue, K. and Aki, S. (2009). Distributions of runs and scans on higher-order Markov trees. Commun. Statist. Theory Meth. 38, 621–641.CrossRefGoogle Scholar
Kochar, S., Mukerjee, H. and Samaniego, F.J. (1999). The ‘signature’ of a coherent system and its application to comparison among systems. Naval Res. Logistics 46, 507–523.Google Scholar
Kuo, W. and Zuo, M. J. (2003). Optimal Reliability Modeling: Principles and Applications. John Wiley, Hoboken, NJ.Google Scholar
Makri, F. S. and Psillakis, Z. M. (2010). On success runs of length exceeded a threshold. To appear in Methodology Comput. Appl. Prob. Google Scholar
Makri, F. S., Philippou, A. N. and Psillakis, Z. M. (2007a). Shortest and longest length of success runs in binary sequences. J. Statist. Planning Infer. 137, 2226–2239.Google Scholar
Makri, F. S., Philippou, A. N. and Psillakis, Z. M. (2007b). Success run statistics defined on an urn model, Adv. Appl. Prob. 39, 991–1019.Google Scholar
Naus, J. (1974). Probabilities for a generalized birthday problem. J. Amer. Statist. Assoc. 69, 810–815.Google Scholar
Naus, J. I. (1982). Approximations for distributions of scan statistics. J. Amer. Statist. Assoc. 77, 177–183.Google Scholar
Navarro, J. and Rychlik, T. (2007). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113.Google Scholar
Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72.Google Scholar
Samaniego, F. J. (2007). System signatures and their applications in engineering reliability. Springer, New York.Google Scholar
Saperstein, B. (1972). The generalized birthday problem. {J. Amer. Statist. Assoc.} 67, 425–428.Google Scholar
Tong, Y. L. (1985). A rearrangement inequality for the longest run, with an application to network reliability. J. Appl. Prob. 22, 386–393.Google Scholar
Zhenkiu, Z. and Glaz, J. (2008). Bayesian variable window scan statistics. J. Statist. Planning Infer. 138, 3561–3567.Google Scholar