Tail index estimation based on survey data

¹ MODAL’X - Université Paris Ouest, 92001 Nanterre, France
² Laboratoire de Statistique, CREST, France
³ Laboratoire AGM - Université de Cergy-Pontoise, 95000 Cergy-Pontoise, France
⁴ Institut Mines-Télécom - LTCI UMR Télécom ParisTech/CNRS No. 5141, 75634 Paris, France.
stephan.clemencon@telecom-paristech.fr

Received: 12 August 2013
Revised: 5 February 2014

Abstract

This paper is devoted to tail index estimation in the context of survey data. Assuming that the population of interest is described by a heavy-tailed statistical model, we prove that the survey scheme plays a crucial role in the design of consistent inference methods for extremes. As can be revealed by simulation experiments, ignoring the sampling plan generally induces a significant bias, jeopardizing the accuracy of the extreme value statistics thus computed. Focus is here on the celebrated Hill method for tail index estimation, it is shown how to modify it in order to take into account the survey design. Precisely, under specific conditions on the inclusion probabilities of first and second orders, we establish the consistency of the variant of the Hill estimator we propose. Additionally, its asymptotic normality is proved in a specific situation. Application of this limit result for building Gaussian confidence intervals is thoroughly discussed and illustrated by numerical results.