On the estimation of the functional Weibull tail-coefficient

We present a nonparametric family of estimators for the tail index of a Weibull tail-distribution when functional covariate is available. Our estimators are based on a kernel estimator of extreme conditional quantiles. Asymptotic normality of the estimators is proved under mild regularity conditions. Their finite sample performances are illustrated both on simulated and real data.

[1]  S. Girard,et al.  Weibull tail-distributions revisited: a new look at some tail estimators , 2011 .

[2]  J. Teugels,et al.  Practical Analysis of Extreme Values , 1996 .

[3]  Nadia L. Kudraszow,et al.  Uniform consistency of kNN regressors for functional variables , 2013 .

[4]  Alexandru V. Asimit,et al.  Pitfalls in Using Weibull Tailed Distributions , 2009 .

[5]  J. Beirlant,et al.  A new estimation method for Weibull-type tails based on the mean excess function , 2009 .

[6]  P. Vieu,et al.  Nonparametric regression for functional data: automatic smoothing parameter selection , 2007 .

[7]  M. Samanta,et al.  Non-parametric estimation of conditional quantiles , 1989 .

[8]  R. Bass.  Regular conditional probabilities , 2011 .

[9]  C. J. Stone,et al.  Consistent Nonparametric Regression , 1977 .

[10]  Stéphane Girard,et al.  Functional nonparametric estimation of conditional extreme quantiles , 2010, J. Multivar. Anal..

[11]  S. Girard,et al.  Functional kernel estimators of large conditional quantiles , 2011, 1107.2261.

[12]  S. Girard.  A Hill Type Estimator of the Weibull Tail-Coefficient , 2004 .

[13]  J. Beirlant,et al.  Semiparametric lower bounds for tail index estimation , 2006 .

[14]  Laurent Gardes,et al.  Comparison of Weibull tail-coefficient estimators , 2011, 1104.0764.

[15]  L. Haan,et al.  Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation , 2000 .

[16]  Laurent Gardes,et al.  Estimation of the Weibull tail-coefficient with linear combination of upper order statistics , 2008, 1103.5894.

[17]  M. Broniatowski.  On the estimation of the Weibull tail coefficient , 1993 .

[18]  S. Girard,et al.  On kernel smoothing for extremal quantile regression , 2012, 1312.5123.

[19]  Enea G. Bongiorno,et al.  Contributions in Infinite-Dimensional Statistics and Related Topics , 2014 .

[20]  Jan Beirlant,et al.  Modeling large claims in non-life insurance , 1992 .

[21]  M. Neves,et al.  Alternatives to a Semi-Parametric Estimator of Parameters of Rare Events—The Jackknife Methodology* , 2000 .

[22]  H. Shang.  Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density , 2014, 1403.1913.

[23]  M. Meerschaert.  Regular Variation in R k , 1988 .

[24]  Winfried Stute,et al.  Conditional empirical processes , 1986 .

[25]  Jonathan El Methni,et al.  Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions , 2014 .

[26]  Frédéric Ferraty,et al.  The Functional Nonparametric Model and Application to Spectrometric Data , 2002, Comput. Stat..

[27]  Aldo Goia,et al.  A partitioned Single Functional Index Model , 2015, Comput. Stat..

[28]  Jonathan El Methni,et al.  Nonparametric estimation of extreme risk measures from conditional heavy-tailed distributions , 2013 .

[29]  J. Diebolt,et al.  Bias-reduced estimators of the Weibull tail-coefficient , 2008, 1103.6172.

[30]  G. Roussas.  Nonparametric Estimation of the Transition Distribution Function of a Markov Process , 1969 .

[31]  Frédéric Ferraty,et al.  Conditional Quantiles for Dependent Functional Data with Application to the Climatic El Niño Phenomenon , 2005 .

[32]  Alain Berlinet,et al.  Asymptotic normality of convergent estimates of conditional quantiles , 2001 .

[33]  J. Teugels,et al.  The mean residual life function at great age: Applications to tail estimation , 1995 .

[34]  S. Girard,et al.  Kernel estimators of extreme level curves , 2011 .

[35]  P. Vieu,et al.  Rate of uniform consistency for nonparametric estimates with functional variables , 2010 .