Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T08:24:50.033Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic results for the multiple scan statistic

Part of: Limit theorems Distribution theory

Published online by Cambridge University Press:  04 April 2017

M. V. Boutsikas ,
M. V. Koutras  and
F. S. Milienos
M. V. Boutsikas*
Affiliation:
University of Piraeus
M. V. Koutras*
Affiliation:
University of Piraeus
F. S. Milienos*
Affiliation:
University of Piraeus
*
* Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 18534 Piraeus, Greece.
* Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 18534 Piraeus, Greece.
* Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 18534 Piraeus, Greece.
Get access Rights & Permissions [Opens in a new window]

Abstract

The contribution of the theory of scan statistics to the study of many real-life applications has been rapidly expanding during the last decades. The multiple scan statistic, defined on a sequence of n Bernoulli trials, enumerates the number of occurrences of k consecutive trials which contain at least r successes among them (r≤k≤n). In this paper we establish some asymptotic results for the distribution of the multiple scan statistic, as n,k,r→∞ and illustrate their accuracy through a simulation study. Our approach is based on an appropriate combination of compound Poisson approximation and random walk theory.

Keywords

Scan statisticcompound Poisson approximationKolmogorov distancetotal variation distanceconvergence in distributionrandom walk

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 320 - 330
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balakrishnan, N. and Koutras, M. V. (2002).Runs and Scans with Applications.John Wiley,New York.Google Scholar
Balakrishnan, N., Koutras, M. V. and Milienos, F. S. (2014).Start-up demonstration tests: models, methods and applications, with some unifications.Appl. Stoch. Models Bus. Ind. 30,373–413.CrossRefGoogle Scholar
Boutsikas, M. V. and Koutras, M. V. (2002).Modeling claim exceedances over thresholds.Insurance Math. Econom. 30,67–83.CrossRefGoogle Scholar
Boutsikas, M. V. and Koutras, M. V. (2003).Bounds for the distribution of two-dimensional binary scan statistics.Prob. Eng. Inform. Sci. 17,509–525.Google Scholar
Boutsikas, M. V. and Koutras, M. V. (2006).On the asymptotic distribution of the discrete scan statistic.J. Appl. Prob. 43,1137–1154.Google Scholar
Boutsikas, M. V., Koutras, M. V. and Milienos, F. S. (2009).Extreme value results for scan statistics. In Scan Statistics (Stat. Ind. Technol.).Birkhäuser,Boston, pp.57–82.Google Scholar
Chen, J. and Glaz, J. (1999).Approximations for the distribution and the moments of discrete scan statistics. In Scan Statistics and Applications (Stat. Ind. Technol.).Birkhäuser,Boston, pp.27–66.Google Scholar
Christofides, T. and Vaggelatou, E. (2004).A connection between supermodular ordering and positive/negative association.J. Multivariate Anal. 88,138–151.Google Scholar
Fu, J. C. (2001).Distribution of the scan statistic for a sequence of bistate trials.J. Appl. Prob. 38,908–916.Google Scholar
Fu, J. C., Wi, T. L. and Lou, W. Y. W. (2012).Continuous, discrete, and conditional scan statistics.J. Appl. Prob. 49,199–209.Google Scholar
Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn.John Wiley,New York.Google Scholar
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
Glaz, J., Pozdnyakov, V. and Wallenstein, S. (eds) (2009).Scan Statistics.Birkhäuser,Boston.Google Scholar
Glaz, J., Naus, J., Roos, M. and Wallenstein, S. (1994).Poisson approximation for the distribution and moments of ordered m-spacings.J. Appl. Prob. 31A,271–281.Google Scholar
Joag-Dev, K. and Proschan, F. (1983).Negative association of random variables, with applications.Ann. Statist. 11,286–295.Google Scholar
Koutras, M. V. (1996).On a waiting time distribution in a sequence of Bernoulli trials.Ann. Inst. Statist. Math. 48,789–806.Google Scholar
Koutras, M. V. and Alexandrou, M. V. (1995).Runs, scans and urn model distributions: a unified Markov chain approach.Ann. Inst. Statist. Math. 47,743–766.Google Scholar
Soon, S. Y. T. (1996).Binomial approximation for dependent indicators.Statist. Sinica. 6,703–714.Google Scholar
Wald, A. and Wolfowitz, J. (1943).An exact test for randomness in the non-parametric case based on serial correlation.Ann. Math. Statist. 14,378–388.Google Scholar
Wu, T. L., Glaz, J. and Fu, J. C. (2013).Discrete, continuous and conditional multiple window scan statistics.J. Appl. Prob. 50,1089–1101.CrossRefGoogle Scholar