Abstract
LetX 1 ... X N be a sequence of independent and identically distributed nonnegative integer valued random variables. For 2 ≤ m ≤ N, consider the moving sums of m consecutive observations. The discrete scan statistic is defined as the maximum value of these moving sums. Conditional on the sum of all the observations, we refer to this scan statistic as the conditional scan statistic.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aldous, D. (1989).Probability Approximations via the Poisson Clumping HeuristicNew York: Springer-Verlag.
Altschul, S. F. and Erickson, B. W. (1988). Significance levels for biological sequence comparison using non-linear similarity functionsBulletin of Mathematical Biology 5077–92.
Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen-Stein methodAnnals of Applied Probability 179–25.
Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen-Stein methodStatistical Science 5403–434.
Arratia, R., Gordon, L. and Waterman, M. (1986). An extreme value theory for sequence matchingAnnals of Statistics 14971–993.
Balakrishnan, N., Balasubramanian, K. and Viveros, R. (1993). On sampling inspection plans based on the theory of runsThe Mathematical Scientist 18113–126.
Balasubramanian, K., Viveros, R. and Balakrishnan, N. (1993). Sooner and later waiting time problems for Markovian Bernoulli trialsStatistics Probability Letters 18153–161.
Banjevic, D. (1990). On order statistics in waiting time for runs in Markov chainsStatistics & Probability Letters 9125–127.
Barbour, A. D., Chryssaphinou, O. and Roos, M. (1995). Compound Poisson approximation in reliability theoryIEEE Transactions on Reliability 44398–402.
Barbour, A. D., Holst, L. and Janson, S. (1992).Poisson ApproximationsOxford, England: Oxford University Press.
Bogush, Jr., A. J. (1972). Correlated clutter and resultant properties of binary signalsIEEE Transactions on Aerospace Electronic Systems 9208–213.
Chao, M. T., Fu, J. C. and Koutras, M. V. (1995). Survey of reliability studies of consecutive-k-out-of-n: F and related systemsIEEE Transactions on Reliability 44120–127.
Chen, J. and Glaz, J. (1995). Two dimensional discrete scan statisticsTechnical Report No. 19Department of Statistics, University of Connecticut, Storrs, CT.
Chen, J. and Glaz, J. (1996). Two dimensional discrete scan statisticsStatistics ε Probability Letters 3159–68.
Chen, J. and Glaz, J. (1997). Approximations and inequalities for the distribution of a scan statistic for 0–1 Bernoulli trials, InAdvances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz. Chapter 16(Eds., N. L. Johnson and N. Balakrishnan), pp. 285–298, New York: John Wiley & Sons.
.Chen, J. and Glaz, J. (1997). Approximation for discrete scan statistics on the circlesubmitted for publication.
Chryssaphinou, O. and Papastavridis, S. G. (1990). Limit distribution for a consecutive-k-out-of-n: F systemAdvances in Applied Probability 22491–493.
Darling, R. W. R. and Waterman, M. S. (1986). Extreme value distributions for the largest cube in random latticeSIAM Journal of Applied Mathematics 46118–132.
Fousler, D. E. and Karlin, S. (1987). Maximal success duration for a semiMarkov processStochastic Processes and their Applications 24203–224.
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n: F system with (k1)-step Markov dependenceIEEE Transactions on Reliability 35602–603.
Fu, J. C. and Hu, B. (1987). On reliability of a large consecutive-k-outof-n: F stystem with (k-1)-step Markov dependenceIEEE Transactions on Reliability 3675–77.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approachJournal of the American Statistical Association 891050–1058.
Fu, J. C. and Koutras, M. V. (1994). Poisson approximation for 2dimensional patternsAnnals of the Institute of Statistical Mathematics 461979–1992.
Fu, Y. X. and Curnow, R. N. (1990). Locating a changed estimation of multiple change pointsBiometrika 77295–304.
Glaz, J. (1983). Moving window detection for discrete dataIEEE Transactions on Information Theory 29457–462.
Glaz, J.(1995). Discrete scan statistics with applications to minefields detection, InProceedings of Conference SPIE 2765 pp. 420–429, Orlando, FL.
Glaz, J. and Naus, J. (1983). Multiple cluster on the lineCommunications in Statistics-Theory and Methods 121961–1986.
Glaz, J. and Naus, J. I. (1991). Tight bounds and approximations for scan statistic probabilities for discrete dataAnnals of Applied Probability 1306–318.
Glaz, J. Naus, J., Roos, M. and Wallenstein, S. (1994). Poisson approximations for the distribution and moments of ordered m-spacingsJournal of Applied Probability 31271–281.
Godbole, A. P. (1990). Specific formulae for some success runs distributionsStatistics é4 Probability Letters 10119–124.
Godbole, A. P. (1991). Poisson approximations for runs and patterns of rare eventsAdvances in Applied Probability 23851–865.
Godbole A. P. (1993). Approximate reliabilities of m-consecutive-k-outof-n failure systemsStatistica Sinica 3321–327.
Goldstein, L. and Waterman, M. S. (1992). Poisson, compound Poisson and process approximations for testing statistical significance in sequence comparisonsBulletin of Mathematical Biology 54785–812.
Gordon, L., Schilling, M. F. and Waterman, M. S. (1986). An extreme value theory for long head runsProbability Theory Related Fields 72279–288.
Gotoh, O. (1990). Optimal sequence alignmentsBulletin of Mathematical Biology 52509–525.
Greenberg, I. (1970). On sums of random variables defined on a two-state Markov chainJournal of Applied Probability 13604–607.
Hirano, K. and Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chainStatistica Sinica 3313–320.
Karlin, S., Blaisdell, B. Mocarski, E. and Brendel, V. (1989). A method to identify distinctive charge configurations in protein sequences with applications to human Herpesvirus polypeptidesJournal of Molecular Biology 205165–177.
Karlin, S. and Ost, F. (1987). Counts of long aligned word matches among random letter sequencesAdvances in Applied Probability 19293–351.
Karwe, V. and Naus, J. (1997). New recursive methods for scan statistic probabilitiesComputational Statistics Data Analysis 23389–404.
Koutras, M. V. and Alexandrou V. A. (1996). Runs, scans and urn model distributions: A unified Markov chain approachAnnals of the Institute of Statistical Mathematics 47743–766.
Koutras, M. V. and Alexandrou V. A. (1997). Non-parametric randomness test based on success runs of fixed lengthStatistics ε Probability Letters 32393–404.
Koutras, M. V. and Papastavridis, S. G. (1993).New Trends in System Reliability EvaluationElsevier Science Publ. B. V. pp. 228–248.
Koutras, M. V., Papadopoulos, G. K. and Papastavridis, S. G. (1993). Reliability of 2-dimensional consecutive-k-out-of-n: F systemsIEEE Transactions on Reliability 42658–661.
Krauth, J. (1992). Bounds for the upper-tail probabilities of the circular ratchet scan statisticBiometrics 481177–1185.
Lou, W. Y. W. (1997). An application of the method of finite Markovchain into runs testsStatistics ε Probability Letters 31155–161.
Mosteller, F. (1941). Note on an application of runs to quality control chartsAnnals of Mathematical Statistics 12228–232.
Mott, R. F., Kirkwood, T. B. L. and Curnow, R. N. (1990). An accurate approximation to the distribution of the length of longest matching word between two random DNA sequencesBulletin of Mathematical Biology 52773–784
Naus, J. L (1974). Probabilities for a generalized birthday problemJournal of the American Statistical Association 69810–815.
Naus, J. I. (1982). Approximations for distributions of scan statisticsJournal of the American Statistical Association 77377–385.
Naus, J. I. and Sheng, K. N. (1996). Screening for unusual matched segments in multiple protein sequencesCommunications in Statistics-Simulation and Computation 25937–952.
Naus, J. I. and Sheng, K. N. (1997). Matching among multiple random sequencesBulletin of Mathematical Biology 59483–496.
Nelson, J. B. (1978). Minimal order models for false alarm calculations on sliding windowsIEEE Transactions on Aerospace and Electronic System 15352–363.
Patefield, W. M. (1981). An efficient method of generating randomR x Ctables with given row and column totalsApplied Statistics 30 91–97.
Philippou, A. N. and Makri, F. S. (1986). Successes, runs and longest runsStatistics e4 Probability Letters 4211–215.
Roos, M. (1993a). Compound Poisson approximations for the number of extreme spacingsAdvances in Applied Probability 25847–874.
Roos, M. (1993b). Stein-Chen Method for compound Poisson ApproximationPh.D. DissertationUniversity of Zurich, Zurich, Switzerland.
Roos, M. (1994). Stein’s method for compound Poisson approximationAnnals of Applied Probability 41177–1187.
Saperstein, B. (1972). The generalized birthday problemJournal of the American Statistical Association 67425–428.
Schwager, S. J. (1983). Run probabilities in sequences of Markov-dependeni trialsJournal of the American Statistical Association 78168–175.
Sheng, K. N. and Naus, J. I. (1994). Pattern matching between two nonaligned random sequencesBulletin of Mathematical Biology 561143–1162.
Sheng, K. N. and Naus, J. I. (1996). Matching rectangles in 2-dimensionsStatistics ei Probability Letters 2683–90.
Viveros, R. and Balakrishnan, N. (1993). Statistical inference from startup demonstration test dataJournal of Quality Technology 25119–130.
Wallenstein, S., Naus, J. and Glaz J. (1994). Power of the scan statistic in detecting a changed segment in a Bernoulli sequenceBiometrika 81595–601.
.Wallenstein, S. and Neff, N. (1987). An approximation for the distribution of the scan statisticStatistics in Medicine 6197–207.
.Wallenstein, S., Weinberg, C. R. and Gould, M. (1989). Testing for a pulse in seasonal event dataBiometrics 45817–830.
Waterman, M. S. (1995).Introduction to Computational BiologyLondon, England: Chapman&Hall.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media New York
About this chapter
Cite this chapter
Chen, J., Glaz, J. (1999). Approximations for the Distribution and the Moments of Discrete Scan Statistics. In: Glaz, J., Balakrishnan, N. (eds) Scan Statistics and Applications. Statistics for Industry and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1578-3_2
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1578-3_2
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7201-4
Online ISBN: 978-1-4612-1578-3
eBook Packages: Springer Book Archive