Testing tail monotonicity by constrained copula estimation

In this paper the interest is in testing for tail monotonicity dependence structures between two random variables. The main focus in the presentation of the statistical methodology is on left tail decreasingness, but the developed procedures can also be used for testing for other specific tail monotonicity dependence structures. In order to assess the p-values of the test statistic, we resample from a constrained copula estimator. This can be done in a nonparametric or in a parametric way. The main difficulty is the construction of a constrained estimator and the development of a resampling technique. The finite-sample performances of the proposed testing procedures are investigated in a simulation study and illustrations on real data examples are provided.

[1]  Michel Denuit,et al.  Actuarial Theory for Dependent Risks: Measures, Orders and Models , 2005 .

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

[3]  A. McNeil.  Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory , 1997, ASTIN Bulletin.

[4]  A study on LTD and RTI positive dependence orderings , 2008 .

[5]  M. J. Frank.  On the simultaneous associativity ofF(x,y) andx +y -F(x,y) , 1979 .

[6]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[7]  John W. Tukey,et al.  A Problem of Berkson, and Minimum Variance Orderly Estimators , 1958 .

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

[9]  I. Gijbels,et al.  Positive quadrant dependence tests for copulas , 2010 .

[10]  Michel Denuit,et al.  Nonparametric Tests for Positive Quadrant Dependence , 2002 .

[11]  Moshe Shaked,et al.  Some positive dependence stochastic orders , 2006 .

[12]  Moshe Shaked,et al.  Some notions of multivariate positive dependence , 2005 .

[13]  Projection estimators of Pickands dependence functions , 2008 .

[14]  I. Gijbels,et al.  Positive quadrant dependence testing and constrained copula estimation , 2013 .

[15]  V. Yohai,et al.  Robust location estimation with missing data , 2010, 1004.5418.

[16]  O. Scaillet.  Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall , 2004 .

[17]  Jan Dhaene,et al.  Risk Measures and Comonotonicity: A Review , 2006, Stochastic Models.

[18]  M. J. Frank.  On the simultaneous associativity ofF(x, y) andx+y−F(x, y) , 1978 .

[19]  Irène Gijbels,et al.  Conditional copulas, association measures and their applications , 2011, Comput. Stat. Data Anal..

[20]  Hélène Cossette,et al.  TVaR-based capital allocation with copulas , 2009 .

[21]  Mary C. Meyer.  A Simple New Algorithm for Quadratic Programming with Applications in Statistics , 2013, Commun. Stat. Simul. Comput..

[22]  C. Genest.  Frank's family of bivariate distributions , 1987 .

[23]  PAUL EMBRECHTS,et al.  Modelling of extremal events in insurance and finance , 1994, Math. Methods Oper. Res..

[24]  O. Scaillet,et al.  A Kolmogorov-Smirnov Type Test for Positive Quadrant Dependence , 2005 .

[25]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[26]  I. Gijbels,et al.  Estimation of a Conditional Copula and Association Measures , 2011 .

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

[28]  E. Lehmann,et al.  Testing Statistical Hypothesis. , 1960 .

[29]  Peter Blum,et al.  The Art of Dependence Modelling: The Latest Advances in Correlation Analysis , 2001 .

[30]  J. D. Esary,et al.  Relationships Among Some Concepts of Bivariate Dependence , 1972 .