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New maximal inequalities for N-demimartingales with scan statistic applications

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  22 June 2017

Markos V. Koutras  and
Demetrios P. Lyberopoulos
Markos V. Koutras*
Affiliation:
University of Piraeus
Demetrios P. Lyberopoulos*
Affiliation:
University of Piraeus
*
* Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 185 34 Piraeus, Greece.
* Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 185 34 Piraeus, Greece.
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Abstract

In the present work, some new maximal inequalities for nonnegative N-demi(super)martingales are first developed. As an application, new bounds for the cumulative distribution function of the waiting time for the first occurrence of a scan statistic in a sequence of independent and identically distributed (i.i.d.) binary trials are obtained. A numerical study is also carried out for investigating the behavior of the new bounds.

Keywords

N-demimartingaleN-demisupermartingaledemimartingalescan statisticdemisubmartingaleboundi.i.d. binary trials

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G48: Generalizations of martingales
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 363 - 378
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York. Google Scholar
[2] Bersimis, S., Koutras, M. V. and Papadopoulos, G. K. (2014). Waiting time for an almost perfect run and applications in statistical process control. Methodology Comput. Appl. Prob. 16, 207–222. CrossRefGoogle Scholar
[3] Boutsikas, M. V. and Koutras, M. V. (2002). Modeling claim exceedances over thresholds. Insurance Math. Econom. 30, 67–83. CrossRefGoogle Scholar
[4] 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
[5] Chen, J. and Glaz, J. (1999). Approximations for the distribution and the moments of discrete scan statistics. In Scan Statistics and Applications, eds. J. Glaz and N. Balakrishnan, Birkhäuser, Boston, MA, pp. 27–66. Google Scholar
[6] Chen, C. and Karlin, S. (2007). r-scans statistics of a Poisson process with events transformed by duplications, deletions, and displacements. Anal. Appl. Prob. 39, 799–825. Google Scholar
[7] Christofides, T. C. (2003). Maximal inequalities for N-demimartingales. Arch. Inequal. Appl. 1, 387–397. Google Scholar
[8] Dai, P., Shen, Y., Hu, S. and Yang, W. (2014). Some results for demimartingales and N-demimartingales. J. Inequal. Appl. 2014:489, 12pp. Google Scholar
[9] Fu, J. C. (2001). Distribution of the scan statistic for a sequence of bistate trials. J. Appl. Prob. 38, 908–916. Google Scholar
[10] Fu, J. C., Wu, T. L. and Lou, W. Y. W. (2012). Continuous, discrete, and conditional scan statistics. J. Appl. Prob. 49, 199–209. CrossRefGoogle Scholar
[11] 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
[12] Glaz, J., Pozdnyakov, V. and Wallenstein, S. (2009). Scan Statistics: Methods and Applications. Birkhäuser, Boston. CrossRefGoogle Scholar
[13] Hu, S. H., Yang, W. Z., Wang, X. J. and Shen, Y. (2010). A note on the inequalities for N-demimartingales and demimartingales. J. Systems Sci. Math. Sci. 30, 1052–1058 (in Chinese). Google Scholar
[14] Newman, C. M. and Wright, A. L. (1982). Associated random variables and martingale inequalities. Z. Wahrscheinlichkeitsth. 59, 361–371. CrossRefGoogle Scholar
[15] Papastavridis, S. G. and Koutras, M. V. (1993). Bounds for reliability of consecutive k-within-m-out-of-n:F systems. IEEE Trans. Reliab. 42, 156–160. CrossRefGoogle Scholar
[16] Pozdnyakov, V. and Steele, J. M. (2009). Martingale methods for patterns and scan statistics. In Scan Statistics: Methods and Applications, eds. J. Glaz et al., Birkhäuser, Boston, MA, pp. 289–317. CrossRefGoogle Scholar
[17] Pozdnyakov, V., Glaz, J., Kulldorff, M. and Steele, J. M. (2005). A martingale approach to scan statistics. Ann. Inst. Statist. Math. 57, 21–37. Google Scholar
[18] Prakasa Rao, B. L. S. (2012). Associated Sequences, Demimartingales and Nonparametric Inference. Birkhäuser, Springer, Basel. Google Scholar
[19] 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