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Discrete, Continuous and Conditional Multiple Window Scan Statistics

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  30 January 2018

Tung-Lung Wu ,
Joseph Glaz  and
James C. Fu
Tung-Lung Wu*
Affiliation:
University of Connecticut
Joseph Glaz*
Affiliation:
University of Connecticut
James C. Fu*
Affiliation:
University of Manitoba
*
∗ Postal address: Department of Statistics, University of Connecticut, Storrs, CT 06269-4120, USA.
∗ Postal address: Department of Statistics, University of Connecticut, Storrs, CT 06269-4120, USA.
∗∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada. Email address: [email protected]
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Abstract

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The distributions of discrete, continuous and conditional multiple window scan statistics are studied. The finite Markov chain imbedding technique has been applied to obtain the distributions of fixed window scan statistics defined from a sequence of Bernoulli trials. In this manuscript the technique is extended to compute the distributions of multiple window scan statistics and the exact powers for multiple pulse and Markov dependent alternatives. An application in blood component quality monitoring is provided. Numerical results are also given to illustrate our theoretical results.

Keywords

Scan statisticmultiple windowfinite Markov chain imbeddingrandom permutationPoisson processpower

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1089 - 1101
Copyright
© Applied Probability Trust 

References

Chen, J. and Glaz, J. (2009). Approximations for two-dimensional variable window scan statistics. In Scan Statistics, eds Glaz, J., Pozdnyakov, V. and Wallenstein, S., Birkhäuser, Boston, MA, pp. 109–128.Google Scholar
Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley, New York.Google Scholar
Cucala, L. (2008). A hypothesis-free multiple scan statistic with variable window. Biometrical J. 50, 299–310.Google Scholar
Fu, J. C. (2001). Distribution of the scan statistic for a sequence of bistate trials. J. Appl. Prob. 38, 908–916.CrossRefGoogle Scholar
Fu, J. C. and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications. World Scientific Publishing Co. Inc., River Edge, NJ.CrossRefGoogle Scholar
Fu, J. C., Wu, T.-L. and Lou, W. Y. W. (2012). Continuous, discrete, and conditional scan statistics. J. Appl. Prob. 49, 199–209.Google Scholar
Glaz, J. (1989). Approximations and bounds for the distribution of the scan statistic. J. Amer. Statist. Assoc. 84, 560–566.CrossRefGoogle Scholar
Glaz, J. and Zhang, Z. (2004). Multiple window discrete scan statistics. J. Appl. Statist. 31, 967–980.Google Scholar
Glaz, J., Pozdnyakov, V. and Wallenstein, S. (eds) (2009). Scan Statistics. Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Haiman, G. (2000). Estimating the distributions of scan statistics with high precision. Extremes 3, 349–361.CrossRefGoogle Scholar
Hoh, J. and Ott, J. ({2000}). {Scan statistics to scan markers for susceptibility genes}. Proc. Nat. Acad. Sci. USA 97, {9615–9617}.CrossRefGoogle ScholarPubMed
Janson, S. (1984). Bounds on the distributions of extremal values of a scanning process. Stoch. Process. Appl. 18, 313–328.Google Scholar
Koutras, M., Papadopoulos, G. and Papastavridis, S. ({1993}). {Reliability of 2-dimensional consecutive-k-out-of-n: F systems}. {IEEE Trans. Reliab.} {42}, {658–661}.Google Scholar
Lachenbruch, P. A., Foulkes, M. A., Williams, A. E. and Epstein, J. S. (2005). Potential use of the scan statistic for quality control in blood product manufacturing. J. Biopharmaceutical Statist. 15, 353–366.CrossRefGoogle ScholarPubMed
Nagarwalla, N. (1996). A scan statistic with a variable window. Statist. Med. 15, 845–850.3.0.CO;2-X>CrossRefGoogle ScholarPubMed
Naus, J. (1974). Probabilities for a generalized birthday problem. J. Amer. Statist. Assoc. 69, 810–815.Google Scholar
Naus, J. I. and Wallenstein, S. (2004). Multiple window and cluster size scan procedures. Methodology Comput. Appl. Prob. 6, 389–400.CrossRefGoogle Scholar
Rakitzis, A. C. and Antzoulakos, D. L. (2011). Chi-square control charts with runs rules. Methodology Comput. Appl. Prob. 13, 657–669.CrossRefGoogle Scholar
Rosenfeld, A. (1978). Clusters in digital pictures. Inform. and Control 39, 19–34.CrossRefGoogle Scholar
Wallenstein, S. R. and Naus, J. I. (1974). Probabilities for the size of largest clusters and smallest intervals. J. Amer. Statist. Assoc. 69, 690–697.CrossRefGoogle Scholar
Wallenstein, S., Naus, J. and Glaz, J. (1994). Power of the scan statistic in detecting a changed segment in a Bernoulli sequence. Biometrika 81, 595–601.CrossRefGoogle Scholar