For heavy tails, classical extreme value index estimators, like the Hill estimator, are usually asymptotically biased. Consequently those estimators are quite sensitive to the number of top order statistics used in the estimation. The recent minimum-variance reduced-bias extreme value index estimators enable us to remove the dominant component of asymptotic bias and keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator. In this paper a new minimum-variance reduced-bias extreme value index estimator is introduced, and its non degenerate asymptotic behaviour is studied. A comparison with another important minimum-variance reduced-bias extreme value index estimator is also provided.
REFERENCES
1.
J.
Beirlant
, F.
Caeiro
and M.I.
Gomes
, An overview and open research topics in statistics of univariate extremes
, Revstat
10
(1
), 1
–31
(2012
).2.
F.
Caeiro
and M.I.
Gomes
, Minimum-variance reduced-bias tail index and high quantile estimation
. Revstat
6
, 1
–20
(2008
).3.
F.
Caeiro
and M.I.
Gomes
, Asymptotic comparison at optimal levels of reduced-bias extreme value index estimators
. Statistica Neerlandica
65
, 462
–488
(2011
).4.
F.
Caeiro
and M.I.
Gomes
, A reduced bias estimator of a ‘scale’ second order parameter. In T. E.
Simos
, G.
Psihoyios
, Ch.
Tsitouras
and Z.
Anastassi
(eds.), AIP Conf. Proc.
1479
, 1114
–1117
(2012
).5.
F.
Caeiro
and M.I.
Gomes
, Bias reduction in the estimation of a shape second-order parameter of a heavy tail model
. Journal of Statistical Computation and Simulation
, 85
(17
), 3405
–3419
(2015
).6.
F.
Caeiro
and M.I.
Gomes
, Threshold selection in extreme value analysis. In D.K.
Dey
and J.
Yan
(eds.), Extreme Value Modeling and Risk Analysis Methods and Applications
, 69
–86
Chapman and Hall/CRC
(2016
).7.
F.
Caeiro
and D. Prata
Gomes
, Adaptive estimation of a tail shape second order parameter: A computational comparative study. In Simos
, T.E.
, Kalogiratou
, Z.
and Monovasilis
, T.
(eds.) AIP Conf. Proc.
1702
, 030005
(2015
).8.
F.
Caeiro
, M.I.
Gomes
and D.D.
Pestana
, Direct reduction of bias of the classical Hill estimator
. Revstat
3
(2
), 111
–136
(2005
).9.
G.
Ciuperca
and C.
Mercadier
, Semi-parametric estimation for heavy tailed distributions
. Extremes
13
(1
), 55
–87
(2010
).10.
E.H.
Deme
, L.
Gardes
and S.
Girard
, On the estimation of the second order parameter for heavy-tailed distributions
. Revstat
11
(3
), 277
–299
(2013
).11.
M.I. Fraga
Alves
, M.I.
Gomes
and L.
de Haan
, A new class of semi-parametric estimators of the second order parameter
. Portugaliae Mathematica
60
(1
), 193
–213
(2003
).12.
Y.
Goegebeur
, J.
Beirlant
and T.
de Wet
, Kernel estimators for the second order parameter in extreme value statistics
. J. Statist. Plann. Inference
140
(9
), 2632
–2654
(2010
).13.
M.I.
Gomes
and A.
Guillou
, Extreme value theory and statistics of univariate extremes: a review
. International Statistical Review
, 83
(2
), 263
–292
(2015
).14.
M.I.
Gomes
and M.J.
Martins
, Generalizations of the Hill estimator: asymptotic versus finite sample behaviour
, J. Statist. Plann. Inference
93
, 161
–180
(2001
).15.
M.I.
Gomes
and M.J.
Martins
, “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter
. Extremes
5
(1
), 5
–31
(2002
).16.
M.I.
Gomes
, M.J.
Martins
and M.M.
Neves
, Improving second order reduced-bias extreme value index estimation
, Revstat
5
(2
), 177
–207
(2007
).17.
M.I.
Gomes
, M.F.
Brilhante
, F.
Caeiro
, and D.
Pestana
, A new partially reduced-bias mean-of-order p class of extreme value index estimators
. Computational Statistics & Data Analysis
82
, 223
–237
(2015
).18.
L.
de Haan
and A.
Ferreira
, Extreme Value Theory: An Introduction
. Springer
, New York
(2006
).19.
L.
de Haan
and L.
Peng
, Comparison of extreme value index estimators
. Statistica Neerlandica
52
, 60
–70
(1998
).20.
L.
Henriques-Rodrigues
, M.I.
Gomes
, M.I. Fraga
Alves
and C.
Neves
, PORT-estimation of a shape second-order parameter
. Revstat
12
(3
), 299
–328
(2014
).21.
B.M.
Hill
, A simple general approach to inference about the tail of a distribution
. Ann. Statist.
3
(5
), 1163
–1174
(1975
).
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