Multivariate extreme value copulas with factor and tree dependence structures

Parsimonious extreme value copula models with O(d) parameters for d observed variables of extrema are presented. These models utilize the dependence characteristics, including factor and tree structures, assumed on the underlying variables that give rise to the data of extremes. For factor structures, a class of parametric models is obtained by taking the extreme value limit of factor copulas with non-zero tail dependence. An alternative model suitable for both factor and tree structures imposes constraints on the parametric Hüsler-Reiss copula to get representations in terms of O(d) other parameters. Dependence properties are discussed. As the full density is often intractable, the method of composite (pairwise) likelihood is used for model inference. Procedures to improve the stability of bivariate density evaluation are also developed. The proposed models are applied to two data examples — one for annual extreme river flows and one for bimonthly extremes of daily stock returns.

[1]  Janos Galambos,et al.  Order Statistics of Samples from Multivariate Distributions , 1975 .

[2]  Aristidis K. Nikoloulopoulos,et al.  Tail dependence functions and vine copulas , 2010, J. Multivar. Anal..

[3]  Kanti V. Mardia,et al.  Multivariate Pareto Distributions , 1962 .

[4]  A. McNeil,et al.  The t Copula and Related Copulas , 2005 .

[5]  M. E. Johnson,et al.  A Family of Distributions for Modelling Non‐Elliptically Symmetric Multivariate Data , 1981 .

[6]  N. Reid,et al.  AN OVERVIEW OF COMPOSITE LIKELIHOOD METHODS , 2011 .

[7]  Richard L. Smith.  Risk Management: Measuring Risk with Extreme Value Theory , 2002 .

[8]  R. Tsay.  Analysis of Financial Time Series: Tsay/Financial Time Series 3E , 2010 .

[9]  H. Joe,et al.  Composite likelihood estimation in multivariate data analysis , 2005 .

[10]  Simone A. Padoan,et al.  Multivariate extreme models based on underlying skew-t and skew-normal distributions , 2011, J. Multivar. Anal..

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

[12]  A. Frigessi,et al.  Pair-copula constructions of multiple dependence , 2009 .

[13]  Aristidis K. Nikoloulopoulos,et al.  Extreme value properties of multivariate t copulas , 2009 .

[14]  G. Maddala,et al.  A Function for Size Distribution of Incomes , 1976 .

[15]  S. Coles,et al.  An Introduction to Statistical Modeling of Extreme Values , 2001 .

[16]  H. Joe.  Dependence Modeling with Copulas , 2014 .

[17]  C. Czado,et al.  Truncated regular vines in high dimensions with application to financial data , 2012 .

[18]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[19]  I. W. Burr.  Cumulative Frequency Functions , 1942 .

[20]  J. Hüsler.  Maxima of normal random vectors: between independence and complete dependence , 1989 .

[21]  Roger M. Cooke,et al.  Probability Density Decomposition for Conditionally Dependent Random Variables Modeled by Vines , 2001, Annals of Mathematics and Artificial Intelligence.

[22]  Mathieu Ribatet,et al.  Spatial extremes: Max-stable processes at work , 2013 .

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

[24]  H. Drees,et al.  Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function , 1998 .

[25]  S. Padoan,et al.  Likelihood-Based Inference for Max-Stable Processes , 2009, 0902.3060.

[26]  D. Clayton.  A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence , 1978 .

[27]  C. Varin,et al.  A note on composite likelihood inference and model selection , 2005 .

[28]  M. Carlson,et al.  A Brief History of the 1987 Stock Market Crash With a Discussion of the Federal Reserve Response , 2006 .

[29]  Marc G. Genton,et al.  On the likelihood function of Gaussian max-stable processes , 2011 .

[30]  D. Cox,et al.  A note on pseudolikelihood constructed from marginal densities , 2004 .

[31]  Harry Joe,et al.  Composite Likelihood Methods , 2012 .

[32]  Pavel Krupskii,et al.  Factor copula models for multivariate data , 2013, J. Multivar. Anal..

[33]  A. Davison,et al.  Statistical Modeling of Spatial Extremes , 2012, 1208.3378.

[34]  Léo R. Belzile,et al.  Multivariate Extreme Value Distributions , 2015 .

[35]  V. P. Godambe.  An Optimum Property of Regular Maximum Likelihood Estimation , 1960 .

[36]  P. Song,et al.  Composite Likelihood Bayesian Information Criteria for Model Selection in High-Dimensional Data , 2010 .

[37]  T. Bedford,et al.  Vines: A new graphical model for dependent random variables , 2002 .

[38]  Harry Joe,et al.  Parametric families of multivariate distributions with given margins , 1993 .

[39]  A. Davison,et al.  Composite likelihood estimation for the Brown–Resnick process , 2013 .

[40]  Koiti Takahasi,et al.  Note on the multivariate burr’s distribution , 1965 .

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