Location Invariant Pickands-type Estimator And Asymptotic Properties Of The Endpoint Estimators

Suppose {X_n, n ≥ 1} be an i.i.d random sequence with common distribution function F(x) and its order statistics is X_1,n ≤ X_2,n ≤ … ≤ X_n,n. If there exist constants a_n > 0, b_n ∈ R and nondegenerate distribution function G(x), such thatthen G(x) must bewe say that F(x) belongs to the domain of attraction of G_γ(x) and write F ∈ D(G_γ),γ is called the extreme value index.In the first part of this article,I propose a location invariant Pickands-type estimatorwhose weak and strong convergence are proved. Asymptotical representation and strong convergence rate are derived, and the optimal choice of k₂ is found. At last simulation and comparasion of this new kind of Pickands-type estimator with other known Pickands-type estimators are considered by using the adaptive algorithm and the MATLAB (mathematics software) programmes.In the second part, I extend the endpoint estimator and propose other two kinds of endpoint estimators, whose weak and strong convergence are proved and the strong convergence rate is derived. Under second regularly varing function condition, the asymptotic normality ofis proved by controling the convergence rate of regularly varing function. Moreover, the results may construct an asymptotic confidence interval for the endpoint x*.