Nonparametric estimator of the tail dependence coefficient: balancing bias and variance

A theoretical expression is derived for the mean squared error of a nonparametric estimator of the tail dependence coefficient, depending on a threshold that defines which rank delimits the tails of a distribution. We propose a new method to optimally select this threshold. It combines the theoretical mean squared error of the estimator with a parametric estimation of the copula linking observations in the tails. Using simulations, we compare this semiparametric method with other approaches proposed in the literature, including the plateau-finding algorithm.

[1]  I. Gijbels,et al.  Choice of smoothing parameter in multivariate copula-based tail coefficients , 2022, Journal of Statistical Planning and Inference.

[2]  Hendrik Supper,et al.  A comparison of tail dependence estimators , 2020, Eur. J. Oper. Res..

[3]  Raphael Huser,et al.  Local Likelihood Estimation of Complex Tail Dependence Structures, Applied to U.S. Precipitation Extremes , 2017, Journal of the American Statistical Association.

[4]  Matthieu Garcin,et al.  Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates , 2017 .

[5]  Paola Zuccolotto,et al.  Dynamic tail dependence clustering of financial time series , 2017 .

[6]  Jennifer L. Wadsworth,et al.  Modeling Spatial Processes with Unknown Extremal Dependence Class , 2017, Journal of the American Statistical Association.

[7]  Matthieu Garcin,et al.  Non-parametric news impact curve: a variational approach , 2017, Soft Computing.

[8]  D. Guégan,et al.  Optimal wavelet shrinkage of a noisy dynamical system with non-linear noise impact , 2015 .

[9]  Chi Xie,et al.  Tail dependence structure of the foreign exchange market: A network view , 2016, Expert Syst. Appl..

[10]  G. McLachlan,et al.  Advances in Data Analysis and Classification , 2015 .

[11]  P. Zuccolotto,et al.  Dynamic tail dependence clustering of financial time series , 2015, Statistical Papers.

[12]  Volker Schmidt,et al.  Joint distributions for total lengths of shortest-path trees in telecommunication networks , 2015, Ann. des Télécommunications.

[13]  Giovanni De Luca,et al.  A tail dependence-based dissimilarity measure for financial time series clustering , 2011, Adv. Data Anal. Classif..

[14]  J. Segers.  Asymptotics of empirical copula processes under non-restrictive smoothness assumptions , 2010, 1012.2133.

[15]  T. Takeuchi,et al.  Copula cosmology: Constructing a likelihood function , 2010, 1011.4997.

[16]  E. Habib,et al.  Estimation of tail dependence coefficient in rainfall accumulation fields , 2009 .

[17]  Qingqing Mao,et al.  FROM FINANCE TO COSMOLOGY: THE COPULA OF LARGE-SCALE STRUCTURE , 2009, 0909.5187.

[18]  F. Serinaldi.  Analysis of inter-gauge dependence by Kendall’s τK, upper tail dependence coefficient, and 2-copulas with application to rainfall fields , 2008 .

[19]  P. Embrechts,et al.  Dependence modeling with copulas , 2007 .

[20]  J. Segers,et al.  A Method of Moments Estimator of Tail Dependence , 2007, 0710.2039.

[21]  Anne-Catherine Favre,et al.  Importance of Tail Dependence in Bivariate Frequency Analysis , 2007 .

[22]  C. Genest,et al.  Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask , 2007 .

[23]  C. Klüppelberg,et al.  Estimating the tail dependence function of an elliptical distribution , 2007 .

[24]  Rafael Schmidt,et al.  Non‐parametric Estimation of Tail Dependence , 2006 .

[25]  Olivier Scaillet,et al.  Testing for Equality between Two Copulas , 2006, J. Multivar. Anal..

[26]  Dominique Guégan,et al.  Empirical estimation of tail dependence using copulas: application to Asian markets , 2005 .

[27]  Markus Junker,et al.  Estimating the tail-dependence coefficient: Properties and pitfalls , 2005 .

[28]  Robert A. Lordo,et al.  Nonparametric and Semiparametric Models , 2005, Technometrics.

[29]  B. Rémillard,et al.  Test of independence and randomness based on the empirical copula process , 2004 .

[30]  M. Wegkamp,et al.  Weak Convergence of Empirical Copula Processes , 2004 .

[31]  J. Tawn,et al.  Extreme Value Dependence in Financial Markets: Diagnostics, Models, and Financial Implications , 2004 .

[32]  Mario V. Wüthrich,et al.  Tail Dependence from a Distributional Point of View , 2003 .

[33]  O. Scaillet,et al.  Nonparametric Estimation of Copulas for Time Series , 2002 .

[34]  A. Juri,et al.  Copula convergence theorems for tail events , 2002 .

[35]  Yannick Malevergne,et al.  Testing the Gaussian copula hypothesis for financial assets dependences , 2001, cond-mat/0111310.

[36]  Christian Genest,et al.  On the multivariate probability integral transformation , 2001 .

[37]  F. Longin,et al.  Extreme Correlation of International Equity Markets , 2000 .

[38]  Janet E. Heffernan,et al.  Dependence Measures for Extreme Value Analyses , 1999 .

[39]  Christian Genest,et al.  A nonparametric estimation procedure for bivariate extreme value copulas , 1997 .

[40]  Richard L. Smith,et al.  Markov chain models for threshold exceedances , 1997 .

[41]  V. Koltchinskii.  M-estimation, convexity and quantiles , 1997 .

[42]  Bruno Rémillard,et al.  On Kendall's Process , 1996 .

[43]  P. Chaudhuri.  On a geometric notion of quantiles for multivariate data , 1996 .

[44]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[45]  A. Ledford,et al.  Statistics for near independence in multivariate extreme values , 1996 .

[46]  M. C. Jones,et al.  A Brief Survey of Bandwidth Selection for Density Estimation , 1996 .

[47]  T. Louis,et al.  Inferences on the association parameter in copula models for bivariate survival data. , 1995, Biometrics.

[48]  C. Genest,et al.  A semiparametric estimation procedure of dependence parameters in multivariate families of distributions , 1995 .

[49]  E. T. Olsen,et al.  Copulas and Markov processes , 1992 .

[50]  Harry Joe,et al.  Bivariate Threshold Methods for Extremes , 1992 .

[51]  R. Theodorescu,et al.  Note on the spatial quantile of a random vector , 1992 .

[52]  J. Angus.  The Asymptotic Theory of Extreme Order Statistics , 1990 .

[53]  C. Genest,et al.  A characterization of gumbel's family of extreme value distributions , 1989 .

[54]  Jonathan A. Tawn,et al.  Bivariate extreme value theory: Models and estimation , 1988 .

[55]  統計数理研究所.  Annals of the institute of statistical mathematics , 1988, Public Choice.

[56]  B. Schweizer,et al.  On Nonparametric Measures of Dependence for Random Variables , 1981 .

[57]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[58]  Masaaki Sibuya,et al.  Bivariate extreme statistics, I , 1960 .

[59]  Y. Hoga.  Structural break tests for extremal dependence in β-mixing random vectors , 2017 .

[60]  M. Haugh,et al.  An Introduction to Copulas , 2016 .

[61]  U. Schepsmeier,et al.  Web supplement: Derivatives and Fisher information of bivariate copulas , 2012 .

[62]  Matthieu,et al.  de Travail du Centre d ’ Economie de la Sorbonne Extreme values of random or chaotic discretization steps , 2012 .

[63]  Андрей Соколов,et al.  Книги издательства Springer Science & Business Media , 2012 .

[64]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[65]  H. A. David.  Order Statistics , 2011, International Encyclopedia of Statistical Science.

[66]  Jean-David Fermanian,et al.  Weak convergence of empirical copula , 2004 .

[67]  Kilani Ghoudi,et al.  Empirical Processes Based on Pseudo-observations 11: The Multivariate Case , 2004 .

[68]  Andrew J. Patton.  On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation , 2002 .

[69]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[70]  H. Joe.  Multivariate Models and Multivariate Dependence Concepts , 1997 .

[71]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[72]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .