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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.
Bertail, Patrice 1, 2 ; Chautru, Emilie 3 ; Clémençon, Stéphan 4
@article{PS_2015__19__28_0,
author = {Bertail, Patrice and Chautru, Emilie and Cl\'emen\c{c}on, St\'ephan},
title = {Tail index estimation based on survey data},
journal = {ESAIM: Probability and Statistics},
pages = {28--59},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2014011},
mrnumber = {3374868},
zbl = {1351.62019},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2014011/}
}
TY - JOUR AU - Bertail, Patrice AU - Chautru, Emilie AU - Clémençon, Stéphan TI - Tail index estimation based on survey data JO - ESAIM: Probability and Statistics PY - 2015 SP - 28 EP - 59 VL - 19 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps/2014011/ DO - 10.1051/ps/2014011 LA - en ID - PS_2015__19__28_0 ER -
%0 Journal Article %A Bertail, Patrice %A Chautru, Emilie %A Clémençon, Stéphan %T Tail index estimation based on survey data %J ESAIM: Probability and Statistics %D 2015 %P 28-59 %V 19 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps/2014011/ %R 10.1051/ps/2014011 %G en %F PS_2015__19__28_0
Bertail, Patrice; Chautru, Emilie; Clémençon, Stéphan. Tail index estimation based on survey data. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 28-59. doi: 10.1051/ps/2014011
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