Robust nonparametric estimation of the conditional tail dependence coefficient

Abstract We consider robust and nonparametric estimation of the coefficient of tail dependence in presence of random covariates. The estimator is obtained by fitting the extended Pareto distribution locally to properly transformed bivariate observations using the minimum density power divergence criterion. We establish convergence in probability and asymptotic normality of the proposed estimator under some regularity conditions. The finite sample performance is evaluated with a small simulation experiment, and the practical applicability of the method is illustrated on a real dataset of air pollution measurements.

[1]  Armelle Guillou,et al.  An asymptotically unbiased minimum density power divergence estimator for the Pareto-tail index , 2013, J. Multivar. Anal..

[2]  Joel Zinn,et al.  Weighted uniform consistency of kernel density estimators , 2004 .

[3]  Andreas Christmann,et al.  A robust estimator for the tail index of Pareto-type distributions , 2007, Comput. Stat. Data Anal..

[4]  P. Embrechts,et al.  Extremes and Robustness: A Contradiction? , 2006 .

[5]  Liang Peng.  Estimation of the coefficient of tail dependence in bivariate extremes , 1999 .

[6]  Mia Hubert,et al.  A Robust Estimator of the Tail Index Based on an Exponential Regression Model , 2004 .

[7]  P. Hall,et al.  Estimating a tail exponent by modelling departure from a Pareto distribution , 1999 .

[8]  D. Pollard,et al.  $U$-Processes: Rates of Convergence , 1987 .

[9]  M. Gomes,et al.  Bias reduction and explicit semi-parametric estimation of the tail index , 2004 .

[10]  J. Teugels,et al.  Statistics of Extremes , 2004 .

[11]  P. Hall.  On Some Simple Estimates of an Exponent of Regular Variation , 1982 .

[12]  M. C. Jones,et al.  Robust and efficient estimation by minimising a density power divergence , 1998 .

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

[14]  Johan Segers,et al.  Second-order refined peaks-over-threshold modelling for heavy-tailed distributions , 2009, 0901.1518.

[15]  Abhik Ghosh,et al.  Divergence based robust estimation of the tail index through an exponential regression model , 2014, Statistical Methods & Applications.

[16]  A. Guillou,et al.  Local robust and asymptotically unbiased estimation of conditional Pareto-type tails , 2014 .

[17]  A. Guillou,et al.  Robust conditional Weibull-type estimation , 2015 .

[18]  Christian Genest,et al.  ON THE TAIL BEHAVIOR OF SUMS OF DEPENDENT RISKS , 2006 .

[19]  Jonathan A. Tawn,et al.  Modelling Dependence within Joint Tail Regions , 1997 .

[20]  William R. Schucany,et al.  Robust and Efficient Estimation for the Generalized Pareto Distribution , 2004 .

[21]  L. Haan,et al.  Extreme value theory : an introduction , 2006 .

[22]  L. de Haan,et al.  Bivariate tail estimation: dependence in asymptotic independence , 2004 .

[23]  L. Peng.  A PRACTICAL WAY FOR ESTIMATING TAIL DEPENDENCE FUNCTIONS , 2010 .

[24]  J. Beirlant,et al.  Some comments on the estimation of a dependence index in bivariate extreme value statistics , 2002 .

[25]  Sangyeol Lee,et al.  Estimation of a tail index based on minimum density power divergence , 2008 .

[26]  Jan Beirlant,et al.  Tail Index Estimation and an Exponential Regression Model , 1999 .

[27]  J. Beirlant,et al.  Bias-reduced estimators for bivariate tail modelling , 2011 .

[28]  Jon A. Wellner,et al.  Empirical processes indexed by estimated functions , 2007, 0709.1013.

[29]  A. Guillou,et al.  Robust and bias-corrected estimation of the coefficient of tail dependence , 2014 .

[30]  A. Guillou,et al.  Local robust estimation of the Pickands dependence function , 2018, The Annals of Statistics.

[31]  E. Giné,et al.  Rates of strong uniform consistency for multivariate kernel density estimators , 2002 .

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

[33]  H. Joe.  Multivariate models and dependence concepts , 1998 .