Abstract
In this chapter, we investigate properties of the exceedances of levels {u n } by ξi, ξ2,…, i.e. the points i for which ξi, > un, and as consequences, obtain limiting distributional results for the kth largest value among ξ1,…, ξ1. In particular, when the familiar assumption \(n\left( {1 - F\left( {u_n } \right)} \right) \to \tau \left( {0 < \tau < \infty } \right)\) holds (Equation (1.5.1)), it will be shown that the exceedances take on a Poisson character as n becomes large. This leads to the limiting distributions for the kth largest values for any fixed rank k = 1, 2,… (the kth “extreme order statistics”) and to their limiting joint distributions.
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© 1983 Springer-Verlag New York Inc.
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Leadbetter, M.R., Lindgren, G., Rootzén, H. (1983). Exceedances of Levels and kth Largest Maxima. In: Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5449-2_2
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DOI: https://doi.org/10.1007/978-1-4612-5449-2_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-5451-5
Online ISBN: 978-1-4612-5449-2
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