On the Generalized Hill Process for Small Parameters and Applications

Abstract

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 > 1.2010 Mathematics Subject Classifications. Primary 62E20, 62F12, 60F05. Secondary 60B10, 60F17.

Reduced‐bias kernel estimators of a positive extreme value index

“…Indeed, the statistics in and are kernel EVI estimators, of the type , built with the power ( ) and log ( ) kernel functions, respectively, where ω is a tuning parameter and Γ denotes the complete Gamma function, defined by , t >0. Also, the class of estimators in was studied in Gomes and Martins and Gomes et al for any real ω ≥ 1 and in Diop and Lo, Lo et al, and Caeiro et al for any real parameter ω >0. Ratios of differences of the statistics in with ω ≥ 1 were considered in Goegebeur et al and Caeiro and Gomes to estimate high‐order tail parameters.…”

“…respectively, where is a tuning parameter and Γ denotes the complete Gamma function, defined by Γ(t) = ∫ ∞ 0 x t−1 e −x dx, t > 0. Also, the class of estimators in (5) was studied in Gomes and Martins 15 and Gomes et al 16 for any real ≥ 1 and in Diop and Lo, 17 Lo et al, 18 and Caeiro et al 19 for any real parameter > 0. Ratios of differences of the statistics in (5) with ≥ 1 were considered in Goegebeur et al 14 and Caeiro and Gomes 20 to estimate high-order tail parameters.…”

“…and(18), respectively, in the ( , )-plane. As may be seen in the figure, there is a region where botĥ ( ) Hill estimator.…”