Limit laws for the norms of extremal samples

Abstract Denote S n ( p ) = k n − 1 ∑ i = 1 k n log ( X n + 1 − i , n ∕ X n − k n , n ) p , where p > 0 , k n ≤ n is a sequence of integers such that k n → ∞ and k n ∕ n → 0 , and X 1 , n ≤ ⋯ ≤ X n , n are the order statistics of iid random variables X 1 , … , X n with regularly varying upper tail of index 1 ∕ γ . The estimator γ ( n ) = ( S n ( p ) ∕ Γ ( p + 1 ) ) 1 ∕ p is an extension of the Hill estimator. We investigate the asymptotic properties of S n ( p ) and γ ( n ) both for fixed p > 0 and for p = p n → ∞ . We prove consistency for γ ( n ) and limit theorem for γ ( n ) − γ under appropriate assumptions. We obtain both Gaussian and non-Gaussian (stable) limit depending on the growth rate of the power sequence p n . Applied to real data we find that for larger p the estimator is less sensitive to the change in k n than the Hill estimator.