A New Class of Semi-Parametric Estimators of the Second Order Parameter

M.I. Fraga Alves, M. Ivette Gomes and Laurens de Haan

Abstract: The main goal of this paper is to develop, under a semi-parametric context, asymptotically normal estimators of the second order parameter $\rho$, a parameter related to the rate of convergence of maximum values, linearly normalized, towards its limit. Asymptotic normality of such estimators is achieved under a third order condition on the tail $1-F$ of the underlying model $F$, and for suitably large intermediate ranks. The class of estimators introduced is dependent on some control or tuning parameters and has the advantage of providing estimators with stable sample paths, as functions of the number $k$ of top order statistics to be considered, for large values of $k$; such a behaviour makes obviously less important the choice of an optimal $k$. The practical validation of asymptotic results for small finite samples is done by means of simulation techniques in Fréchet and Burr models.

Keywords: extreme value theory; tail inference; semi-parametric estimation; asymptotic properties.

Classification (MSC2000): 60G70, 62G32.; 62G05, 62E20, 65C05.

Full text of the article:

• Compressed PostScript file (220 kilobytes)
• PDF file (404 kilobytes)