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Continuous, Discrete, and Conditional Scan Statistics

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  04 February 2016

James C. Fu ,
Tung-Lung Wu  and
W.Y. Wendy Lou
James C. Fu*
Affiliation:
University of Manitoba
Tung-Lung Wu*
Affiliation:
University of Manitoba
W.Y. Wendy Lou*
Affiliation:
University of Toronto
*
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada.
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada.
∗∗∗ Postal address: Dalla Lana School of Public Health, University of Toronto, Toronto, Ontario, M5T 3M7, Canada.
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Abstract

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The distributions for continuous, discrete, and conditional discrete scan statistics are studied. The approach of finite Markov chain imbedding, which has been applied to random permutations as well as to runs and patterns, is extended to compute the distribution of the conditional discrete scan statistic, defined from a sequence of Bernoulli trials. It is shown that the distribution of the continuous scan statistic induced by a Poisson process defined on (0, 1] is a limiting distribution of weighted distributions of conditional discrete scan statistics. Comparisons of rates of convergence as well as numerical comparisons of various bounds and approximations are provided to illustrate the theoretical results.

Keywords

Scan statisticsMarkov chain imbeddingrandom permutationPoisson process

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 49 , Issue 1 , March 2012 , pp. 199 - 209
Copyright
© Applied Probability Trust 

References

Alm, S. E. (1999). Approximations of the distributions of scan statistics of Poisson processes. In Scan Statistics and Applications, Birkhäuser, Boston, MA, pp. 113–139.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, John Wiley, New York, pp. 285–298.Google Scholar
Fu, J. C. (2001). Distribution of the scan statistics for a sequence of bistate trials. J. Appl. Prob. 38, 908–916.Google Scholar
Glaz, J. (1989). Approximations and bounds for the distribution of the scan statistics. J. Amer. Statist. Assoc. 84, 560–566.Google Scholar
Glaz, J. (1992). Approximations for tail probabilities and moments of the scan statistic. Comput. Statist. Data Anal. 14, 213–227.Google Scholar
Glaz, J. and Balkrishnan, N. (eds) (1999). Scan Statistics and Applications. Birkhäuser, Boston, MA.Google Scholar
Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics. Springer, New York.Google Scholar
Haiman, G. (2000). Estimating the distributions of scan statistics with high precision. Extremes 3, 349–361.Google Scholar
Janson, S. (1984). Bounds on the distributions of extremal values of a scanning process. Stoch. Process. Appl. 18, 313–328.Google Scholar
Karlin, S. and McGregor, J. (1959). Coincidence probabilities. Pacific J. Math. 9, 1141–1164.CrossRefGoogle 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.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
Neff, N. D. and Naus, J. I. (1980). Selected Tables in Mathematical Statistics, Vol. VI, American Mathematical Society, Providence, RI.Google Scholar