Tail Estimation Based on Numbers of Near m-Extremes

  • Samuel Müller


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.


Distribution Function Order Statistic Asymptotic Distribution Independent Random Variable Continuous Distribution 
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© 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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