On the Generalized Hill Process for Small Parameters and Applications

Let X 1 , X 2 ,. . .be a sequence of independent copies (s.i.c) of a real random variable (r.v.) X 1, with distribution function df F(x) = P(X x) and let X 1,n X 2,n • • • X n,n be the order statistics based on the n 1 first of these observations.The following continuous generalized Hill processτ > 0, 1 k n, has been introduced as a continuous family of estimators of the extreme value index, and largely studied for statistical purposes with asymptotic normality results restricted to τ > 1/2.We extend those results to 0 < τ 1/2 and show that asymptotic normality is still valid for τ = 1/2.For 0 < τ < 1/2, we get non Gaussian asymptotic laws which are closely related to the Riemann function ζ (s) = ∑ ∞ n=1 n -s , s > 1.