Tail Estimation Based on Numbers of Near m-Extremes

Let {X-n, n greater than or equal to 1} be a sequence of independent random variables with common continuous distribution function F having finite upper endpoint. A new tail index estimator (γ) over cap (n) is defined based on only two numbers of near m-extremesKn(a(i), m) = # {j : X(n - m + 1:n) - a(i) <X-j less than or equal to X(n - m + 1:n), m greater than or equal to 1,where X-(i:n) denotes the i-th order statistic and a(2) > a(1) > 0. The weak and almost sure convergence of (γ) over cap (n) to the tail index gamma, as well as the asymptotic distribution is given. Moreover, the asymptotic distribution of K-n(a(n), m) for a(n) --> 0 is derived.