Tail estimation based on numbers of near m-extremes

Let {Xn,n⩾1} 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
Kn(ai,m)=\# {j:X(n−m+:1)−ai<Xj⩽X(n−m+1:n)},m⩾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.