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Corrected discrete approximations for the conditional and unconditional distributions of the continuous scan statistic

Part of: Geometric probability and stochastic geometry Distribution theory Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  04 April 2017

Yi-Ching Yao ,
Daniel Wei-Chung Miao  and
Xenos Chang-Shuo Lin
Yi-Ching Yao*
Affiliation:
Academia Sinica
Daniel Wei-Chung Miao*
Affiliation:
National Taiwan University of Science and Technology
Xenos Chang-Shuo Lin*
Affiliation:
Aletheia University
*
* Postal address: Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan, ROC. Email address: yao@stat.sinica.edu.tw
** Postal address: Graduate Institute of Finance, National Taiwan University of Science and Technology, Taipei 106, Taiwan, ROC. Email address: miao@mail.ntust.edu.tw
*** Postal address: Accounting Information Department, Aletheia University, New Taipei City, 25103, Taiwan, ROC. Email address: xenos.lin@gmail.com
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Abstract

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). Using a change of measure argument, we derive the first-order term of the discrete approximation which involves some functionals of the Poisson process. Richardson's extrapolation is then applied to yield a corrected (second-order) approximation. Numerical results are presented to compare various approximations.

Keywords

Poisson processRichardson’s extrapolationMarkov chain embeddingchange of measuresecond-order approximation

MSC classification

Primary: 60E05: Distributions 62E20: Asymptotic distribution theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 304 - 319
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Barton, D. E. and Mallows, C. L. (1965).Some aspects of the random sequence.Ann. Math. Statist. 36,236–260.Google Scholar
[2] Chan, H. P. and Zhang, N. R. (2007).Scan statistics with weighted observations.J. Amer. Statist. Assoc. 102,595–602.CrossRefGoogle Scholar
[3] Chang, J. C., Chen, R. J. and Hwang, F. K. (2001).A minimal-automaton-based algorithm for the reliability of Con(d,k,n) system.Methodol. Comput. Appl. Prob. 3,379–386.CrossRefGoogle Scholar
[4] Fu, J. C. (2001).Distribution of the scan statistic for a sequence of bistate trials.J. Appl. Prob. 38,908–916.Google Scholar
[5] Fu, J. C. and Koutras, M. V. (1994).Distribution theory of runs: a Markov chain approach.J. Amer. Statist. Assoc. 89,1050–1058.Google Scholar
[6] 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
[7] Glaz, J. and Naus, J. I. (1991).Tight bounds and approximations for scan statistic probabilities for discrete data.Ann. Appl. Prob. 1,306–318.Google Scholar
[8] Glaz, J. and Naus, J. I. (2010).Scan statistics. In Methods and Applications of Statistics in the Life and Health Sciences, ed. N. Balakrishnan.John Wiley,New Jersey, pp.733–747.Google Scholar
[9] Glaz, J., Naus, J. and Wallenstein, S. (2001).Scan Statistics.Springer,New York.Google Scholar
[10] Glaz, J., Pozdnyakov, V. and Wallenstein, S. (eds) (2009).Scan Statistics.Birkhäuser,Boston.CrossRefGoogle Scholar
[11] Huffer, F. W. and Lin, C.-T. (1997).Computing the exact distribution of the extremes of sums of consecutive spacings.Comput. Statist. Data Anal. 26,117–132.Google Scholar
[12] Huffer, F. W. and Lin, C.-T. (1999).An approach to computations involving spacings with applications to the scan statistic. In Scan Statistics and Applications, eds J. Glaz and N. Balakrishnan.Birkhäuser,Boston, pp.141–163.Google Scholar
[13] Huntington, R. J. and Naus, J. I. (1975).A simpler expression for Kth nearest neighbor coincidence probabilities.Ann. Prob. 3,894–896.Google Scholar
[14] Hwang, F. K. (1977).A generalization of the Karlin–McGregor theorem on coincidence probabilities and an application to clustering.Ann. Prob. 5,814–817.Google Scholar
[15] Janson, S. (1984).Bounds on the distributions of extremal values of a scanning process.Stoch. Process. Appl. 18,313–328.Google Scholar
[16] Karlin, S. and McGregor, G. (1959).Coincidence probabilities.Pacific J. Math. 9,1141–1164.Google Scholar
[17] Koutras, M. V. and Alexandrou, V. A. (1995).Runs, scans, and urn model distributions: a unified Markov chain approach.Ann. Inst. Statist. Math. 47,743–776.Google Scholar
[18] Loader, C. (1991).Large deviation approximations to the distribution of scan statistics.Adv. Appl. Prob. 23,751–771.Google Scholar
[19] Naus, J. I. (1982).Approximations for distributions of scan statistics.J. Amer. Statist. Assoc. 77,177–183.Google Scholar
[20] Neff, N. D. and Naus, J. I. (1980).Selected Tables in Mathematical Statistics, Vol. VI.American Mathematical Society,Providence.Google Scholar
[21] Siegmund, D. and Yakir, B. (2000).Tail probabilities for the null distribution of scanning statistics.Bernoulli 6,191–213.Google Scholar
[22] Wu, T. L., Glaz, J. and Fu, J. C. (2013).Discrete, continuous and conditional multiple window scan statistics.J. Appl. Prob. 50,1089–1101.Google Scholar
[23] Yao, Y.-C., Miao, D. W.-C. and Lin, X. C.-S. (2017).Corrected discrete approximations for the conditional and unconditional distributions of the continuous scan statistic. Preprint. Available at http://arxiv.org/abs1602.02597.Google Scholar