Tail approximations to the density function in EVT

Abstract

Let X 1, X 2, ...,X n be independent identically distributed random variables with common distribution function F, which is in the max domain of attraction of an extreme value distribution, i.e., there exist sequences a n > 0 and b n ∈ ℝ such that the limit of \(P(a_n^{-1}(\max_{1\leq i\leq n}X_i-b\!_n)\leq x)\) exists. Assume the density function f (of F) exists. We obtain an uniformly weighted approximation to the tail density function f, and an uniformly weighted approximation to the tail density function of \(P(a_n^{-1}(\max_{1\leq i\leq n}X_i-b\!_n)\leq x)\) under some second order condition.