# Tail Estimation Based on Numbers of Near m-Extremes

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## 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 *a*_{2} > *a*_{1} > 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 *K*_{n}*(a*_{n}*, m)* for *a*_{n}*→ 0* is derived.

## Keywords

Distribution Function Order Statistic Asymptotic Distribution Independent Random Variable Continuous Distribution## Preview

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