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Extreme Value Results for Scan Statistics

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Scan Statistics

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

In the first part of this chapter we focus on the classical scan and multiple scan statistic, defined on a sequence of independent and identically distributed (i.i.d.) binary trials and review a number of bounds and approximations for their distributions which have been developed by the aid of distance measures. Moreover, we discuss briefly a number of asymptotic results that have been established by setting up appropriate conditions guaranteeing the convergence (to zero) of the distance measures’ upper bounds. In the second part, we study a multiple scan statistic enumerating variable by considering a general threshold-based framework, defined on i.i.d. continuous random variables. More specifically, we first prove a compound Poisson approximation for the total number of fixed length overlapping moving windows containing a prespecified number of threshold exceedances. The classical scan and multiple scan statistic may be treated as a special case of this general model. Next we exploit the previous result to gain some new extreme value results for the scan enumerating statistic under the assumption that the continuous random variables belong to the maximum domain of attraction of one of the three extreme value distributions (Fréchet, reversed Weibull, Gumbel). Finally, we elucidate how the general results can be applied in a number of classical continuous distributions (Pareto, uniform, exponential and normal).

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Notes

  1. 1.

    Products of the form \(\prod _{i={i}_{1}}^{{i}_{2}}f(i)\) with i 1>i 2 are conventionally set equal to 1.

  2. 2.

    Since \(\frac{d} {d\theta }H(\theta ,p)\) =h(θ,p) > 0, the quantity H(θ,p) varies monotonically from 0 to − ln p; therefore, the equation H(θ,p) = c admits a unique solution θ ∈ (p, 1). for 0 < c < −ln p.

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Author information

Authors and Affiliations

  1. Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece

    Michael V. Boutsikas, Markos V. Koutras & Fotios S. Milienos

Authors
  1. Michael V. Boutsikas
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  2. Markos V. Koutras
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  3. Fotios S. Milienos
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Editor information

Editors and Affiliations

  1. Dept. Statistics, University of Connecticut, Glenbrook Rd. 215, Storrs, 06269-3120, U.S.A.

    Joseph Glaz

  2. Dept. Statistics, University of Connecticut, Glenbrook Road 215, Storrs, 06269-4120, U.S.A.

    Vladimir Pozdnyakov

  3. Center for Biomathematical Sciences, Mount Sinai School of Medicine, Gustave L Levy Place 1, New York, 10029, U.S.A.

    Sylvan Wallenstein

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© 2009 Birkhäuser Boston, a part of Springer Science+Business Media, LLC

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Boutsikas, M.V., Koutras, M.V., Milienos, F.S. (2009). Extreme Value Results for Scan Statistics. In: Glaz, J., Pozdnyakov, V., Wallenstein, S. (eds) Scan Statistics. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4749-0_3

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