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Tail Estimation Based on Numbers of Near m-Extremes

  • Samuel Müller
Article

Abstract

Let \(\left\{ {{\text{X}}_n ,n \geqslant 1} \right\}\) be a sequence of independent random variables with common continuous distribution function F having finite upper endpoint. A new tail index estimator γ^ n is defined based on only two numbers of near m-extremes \(K_n \left( {a_i ,m} \right) = {\text{\# }}\left\{ {{\text{j:X}}_{\left( {n - m + :1} \right)} - a_i < X_j \leqslant X_{\left( {n - m + 1:n} \right)} } \right\},m \geqslant 1\) where X (i:n) denotes the -th order statistic and a2 > a1 > 0. The weak and almost sure convergence of γ^ n to the tail index γ, as well as the asymptotic distribution is given. Moreover, the asymptotic distribution of Kn(an, m) for an→ 0 is derived.

Keywords

Distribution Function Order Statistic Asymptotic Distribution Independent Random Variable Continuous Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Samuel Müller
    • 1
  1. 1.Department of Mathematical Statistics and Actuarial ScienceUniversity of BernBernSwitzerland

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