ESTIMATION OF THE PICKANDS DEPENDENCE FUNCTION FOR BIVARIATE ARCHIMAX COPULAS

Transcription

1 ESTIMATION OF THE PICKANDS DEPENDENCE FUNCTION FOR BIVARIATE ARCHIMAX COPULAS SIMON CHATELAIN A THESIS SUBMITTED TO MCGILL UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MCGILL UNIVERSITY MONTRÉAL, QUÉBEC JUNE 215 COPYRIGHT BY SIMON CHATELAIN, 215

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3 Acknowledgements I would like to thank Professor Johanna Nešlehová for her supervision throughout my Master s degree. She has brought me to do research in an area that I knew very little of and now enjoy. This eperience has convinced me to pursue my studies in this domain of multivariate etreme analysis and I look forward to working with her in the future. I am also grateful for her financial support through her Natural Sciences and Engineering Research Council of Canada grant. She has also supported my travels to the 42nd Annual Meeting of the Statistical Society of Canada in Toronto as well as the 9th Etreme Value Analysis conference in Ann Arbor, which were both rich eperiences for me. Professor Anne-Laure Fougères welcomed me in the Camille Jordan Institute of Mathematics at the University of Lyon 1 during the Fall semester of 214. I would like to epress my gratitude to her, as it proved to be a fruitful semester. Chapter 4 is the most crucial chapter in this thesis, and was written there thanks to her support and guidance. I am very ecited to have Anne-Laure as a co-supervisor for my Ph.D. Professor Christian Genest was instrumental to my learning eperience during my Master s degree. He was willing to thoroughly answer any question I had regarding his articles, from which this thesis draws much of its inspiration. He also arranged my summer internship at the research institute of Hydro-Québec IREQ in 214, which was generously financed by his Fonds de recherche du Québec -Nature et technologies grant. Christian helped finance my time in France thanks to his Natural Sciences and Engineering Research Council of Canada grant. I would also like to thank my eternal eaminer, Professor David B. Wolfson, whose comments led to several improvements in this thesis. Finally, I am grateful for the support I have received from my family throughout my studies. iii

4 Abstract This M.Sc. thesis contributes to the use of Archima copulas to model bivariate etremes. After a review of Archimedean and etreme-value copulas, Archima copulas are defined and two corresponding estimators of the Pickands dependence function are proposed. These estimators require no knowledge of the margins, but the Archimedean generator is assumed to be known and to have certain properties of regular variation. Within these imposed conditions, the asymptotic behavior of the estimators is established and related to empirical copula processes and limiting Brownian processes. A simulation study is then conducted, using copulas that fulfill the requirements for consistency as well as copulas that do not. Many questions arise from Chapters 4 and 5, clearly showing where more work on the estimation of Archima copulas is needed. iv

5 Résumé Ce mémoire contribue à l utilisation des copules Archima pour modéliser les données etrêmes bi-variées. Après une revue des copules Archimédiennes et ma-stables, les copules Archima sont définies. Deu estimateurs pour la fonction de dépendance de Pickands sont proposés. Ces estimateurs ne requièrent aucun savoir sur les lois marginales, mais on doit supposer la connaissance du générateur Archimédien et imposer des conditions de variation régulière. Le comportement asymptotique de ces estimateurs est établi, en utilisant des outils tels que le processus de copule empirique et les processus Browniens. Une étude de simulation est menée, en utilisant des copules qui remplissent ou non les conditions nécessaires à la convergence des estimateurs. De cette étude il est évident où plus de travail de recherche est requis pour l ajustement des copules Archima. v

6 Contents Abstract Résumé List of figures List of tables iv v vii i 1 Introduction 1 2 Background Material Copulas Sklar s Theorem Empirical Copulas Etreme Value Copulas Pickands Representation Estimation Archimedean Copulas Stochastic Representation Eamples Random Number Generation Regular Variation Archima Copulas Stochastic representation Random Number Generation vi

7 Contents 3.3 Eamples Asymmetric Gumbel-Hougaard Pickands Dependence Function Clayton Family Miture Family Inverse Pareto-Simple Family Gumbel Generator Estimating the Pickands dependence function in Archima Copulas Pickands type estimator for Archima Copulas Regularly Varying Generators Generators of Finite Support CFG type estimator for Archima Copulas Regularly Varying Generators Generators of Finite Support Summary Simulation Study Clayton Family Inverse Pareto-Simple Family Miture Family Gumbel Family Conclusion Findings Future work vii

8 List of Figures 3.1 Asymmetric Gumbel-Hougaard Pickands dependence function plots. Parameters : κ,λ={.5,.5,.3,.7,.1,.9} left to right and β= Scatter plots n= 1 of random draws from a bivariate Archima copula with Clayton Archimedean generator ψ θ with parameters θ = {.2,.9,2} top to bottom, with Gumbel-Hougaard asymmetric Pickands dependence function with parameters κ, λ = {.5,.5,.3,.7,.1,.9} left to right and β= Scatter plots n = 1 of random draws from a bivariate Archima copula with Pareto & Inverse Pareto miture generator ψ α,γ with parameters α={1,3,6} top to bottom and γ=.3, with Gumbel-Hougard asymmetric Pickands dependence function with parameters κ, λ = {.5,.5,.3,.7,.1,.9} left to right and β= Scatter plots n= 1 of random draws from a bivariate Archima copula with Pareto & Inverse Pareto miture generator ψ α,γ with parameters γ= {.3,.6,.9} top to bottom and α=3, with Gumbel-Hougaard asymmetric Pickands tail dependence function with parameters κ, λ = {.5,.5,.3,.7,.1,.9} left to right and β= Scatter plots n= 1 of random draws from a bivariate Archima copula with Inverse Pareto miture generator ψ γ with parameters γ={.3,.9,5} top to bottom and with Gumbel-Hougaard asymmetric Pickands tail dependence function with parameters κ, λ = {.5,.5,.3,.7,.1,.9} left to right and β= viii

9 List of Figures 3.6 Scatter plots n= 1 of random draws from a bivariate Archima copula with Gumbel Archimedean generator ψ θ with parameters θ= {2,5,1} top to bottom, with asymmetric Gumbel Pickands dependence function with parameters κ, λ = {.5,.5,.3,.7,.1,.9} left to right and β = Rank-based Pickands left and Capéraà-Fougères-Genest right type estimates in black of the Pickands dependence function in red for a sample of size n= 1 from a bivariate Archima copula with asymmetric Gumbel Pickands dependence function with parameters β=3 and κ,λ=.9,.9, and a Clayton generator ψ θ with parameters θ= {.1,.2,.5} top to bottom Rank-based Pickands left and Capéraà-Fougères-Genest right type estimates in black of the Pickands dependence function in red for a sample of size n = 1 from a bivariate Archima copula with an asymmetric Gumbel Pickands dependence function with parameters β=3 and κ,λ=.5,.2, and a Clayton generator ψ θ with parameters θ= {.1,.2,.5} top to bottom Mean Integrated Squared Error MISE for rank-based Pickands in orange and Capéraà-Fougères-Genest in blue type estimators for samples of various sizes from a bivariate Archima copula with an asymmetric Gumbel Pickands dependence function with β=3, κ,λ={.5,.5,.3,.7,.1,.9} left to right and a Clayton generator ψ θ with parameters θ.1,.4. Sample sizes n= 1 full lines, n= 25 dashed lines, n= 5 dotted lines. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE for rank-based Pickands in orange and Capéraà-Fougères-Genest in blue type estimators for samples of various sizes from a bivariate Archima copula with Gumbel Pickands dependence function with β=3 and κ,λ={.5,.5,.3,.7,.1,.9} left to right and an Inverse Pareto-Simple generator ψ γ with parameters γ.1,.9. Sample sizes n= 1 full lines, n= 25 dashed lines, n= 5 dotted lines. Number of Monte Carlo replicates N MC = i

10 List of Tables 2.1 Common Bivariate Archimedean Copulas Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Clayton generator ψ θ and an asymmetric Gumbel Pickands dependence function with parameters β=3 and κ,λ=.5,.5 symmetric case. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Clayton generator ψ θ and an asymmetric Gumbel Pickands dependence function with parameters β=3 and κ,λ=.3,.7 mild asymmetry. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Clayton generator ψ θ and an asymmetric Gumbel Pickands dependence function with parameters β=3 and κ, λ =.1,.9 strong asymmetry. Number of Monte Carlo replicates N MC =

11 List of Tables 5.4 Mean Integrated Squared Error MISE 1 for the rank-based Capéraà- Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Clayton generator ψ θ with high parameter values θ {1,...,9} and an asymmetric Gumbel Pickands dependence function with parameter β=3. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with an Inverse Pareto- Simple generator ψ γ with parameters γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β=3 and κ,λ=.5,.5 symmetric case. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from bivariate Archima copulas with an Inverse Pareto-Simple generator ψ γ with parameters γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β = 3 and κ,λ =.3,.7 mild asymmetry. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from bivariate Archima copulas with an Inverse Pareto-Simple generator ψ γ with parameters γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β=3 and κ,λ=.1,.9 strong asymmetry. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Pareto & Inverse Pareto-Simple generator ψ α,γ with parameters α=4 and γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β = 3 and κ,λ=.5,.5 symmetric case. Number of Monte Carlo replicates N MC = i

12 List of Tables 5.9 Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from bivariate Archima copulas with a Pareto & Inverse Pareto-Simple generator ψ α,γ with parameters α=4 and γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β = 3 and κ,λ=.3,.7 mild asymmetry. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from bivariate Archima copulas with an Inverse Pareto-Simple generator ψ α,γ with parameters α=4 and γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β=3 and κ,λ=.1,.9 strong asymmetry. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Pareto & Inverse Pareto-Simple generator ψ α,γ with parameters α=6 and γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β = 3 and κ,λ=.5,.5 symmetric case. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from bivariate Archima copulas with a Pareto & Inverse Pareto-Simple generator ψ α,γ with parameters α=6 and γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β = 3 and κ,λ=.3,.7 mild asymmetry. Number of Monte Carlo replicates N MC = ii

13 List of Tables 5.13 Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from bivariate Archima copulas with an Inverse Pareto-Simple generator ψ α,γ with parameters α=6 and γ {.1,...,.9} and an asymmetric Gumbel dependence function with parameters β=3 and κ,λ=.1,.9 strong asymmetry. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Gumbel generator ψ θ and an asymmetric Gumbel Pickands dependence function with parameters β=3 and κ,λ=.5,.5 symmetric case. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Gumbel generator ψ θ and an asymmetric Gumbel Pickands dependence function with parameters β=3 and κ,λ=.3,.7 mild asymmetry. Number of Monte Carlo replicates N MC = Mean Integrated Squared Error MISE 1 for rank-based Pickands P and Capéraà-Fougères-Genest CFG type estimators for samples of various sizes from a bivariate Archima copulas with a Gumbel generator ψ θ and an asymmetric Gumbel Pickands dependence function with parameters β=3 and κ, λ =.1,.9 strong asymmetry. Number of Monte Carlo replicates N MC = iii

14 1 Introduction Copulas are becoming increasingly popular in multivariate analysis. They provide a complete description of the dependence structure that ties random variables together, and as such are employed in many applications of statistics. For eample, hydrology, meteorology, climatology, finance and economics are sciences that often require the joint modeling of several quantities. There is an etensive and growing literature on copula modeling techniques; book treatments include Joe 214 and Nelsen 26. Inference techniques are summarized, e.g., in articles such as Genest & Favre 27 and Genest & Nešlehová 212. The analysis of etremes is also an area of statistics and probability which is widely used in applications and currently growing. Natural catastrophes such as floods, hurricanes and heat waves can be seen as the occurrence of a random variable s etreme value. The same can be said for large losses in a stock s return. The main difficulty in the study of etremes is the fact that in practice there is very little data to analyze, since these events are by definition rare. This has led to developments in asymptotic theory that captures the tails of distributions as accurately as possible. See Gnedenko 1943 for important results regarding the Generalized Etreme Value GEV family. For the peaks over threshold POT method, see Balkema & de Haan 1974 and Pickands Comprehensive books on etreme-value theory include, for eample, Coles 21, Embrechts et al and de Haan & Ferreira 26. 1

15 Chapter 1. Introduction Combining the previously mentioned areas, multivariate etremes is the subject of this thesis. In the applications already provided, etreme events are more often than not characterized by several variables jointly eceeding a high threshold. A financial crisis is due to several components crashing, and a hurricane occurs for specific temperature, pressure and wind speed conditions. Ma-stable copulas describe the dependence structure of etreme values, and are usually fitted to samples of component-wise maima. It is at this step that statisticians make the so-called leap of faith, where asymptotic theory is used for inference in finite samples. In practice, there are difficulties with fitting etreme-value copulas to etreme data because we find ourselves in a pre-asymptotic setting. Archima copulas, introduced by Capéraà et al. 2, can help fi this problem. As a large class that generalizes both etremevalue copulas and Archimedean copulas, they can provide the etra fleibility needed in applications. This thesis contributes to modeling techniques for multivariate etremes, specifically to Archima copulas. Very little work has been done on inference for these models. Thanks to their construction, inspiration can be drawn from what is known about inference for Archimedean and etreme-value copulas. In this thesis, two new estimators of the so-called Pickands dependence function are proposed in the bivariate setting. These estimators are rank-based and therefore do not require knowledge of the marginal distributions. In a sense, they could be qualified as semi-parametric since they assume the knowledge of the Archimedean generator of the Archima copula. In Chapter 2, important definitions and concepts are reviewed. First, fundamental results such as the Decomposition Theorem of Sklar 1959 are presented. The motivation and definition of etreme-value copulas are presented, where the crucial Pickands dependence function is defined. Two non-parametric estimators from Genest & Segers 29 for this function are presented as well as their limiting distribution. The Archimedean copula class is widely used in dependence modeling due to its fleibility. Important results related to its stochastic representation are given, as well as a random number generation algorithm and eamples. Finally, we go over important notions of regular variation which will be used in Chapter 4. 2

16 By Chapter 3 we have the necessary elements from which we can define Archima copulas. Indeed, this class of copulas is defined by an Archimedean generator and a stable tail dependence function or Pickands dependence function. Many important results due to Charpentier et al. 214 and Capéraà et al. 2 are presented, particularly regarding the stochastic representation. This leads us to a an algorithm that allows us to draw random observations from bivariate Archima copulas. Finally, we conclude this chapter with eamples and plots of data drawn from Archima copulas. These eamples of Archima copulas will be used again in the simulation study in Chapter 5. Two Pickands dependence function estimators for bivariate Archima copulas are proposed and studied in depth in Chapter 4. Assuming the knowledge of the Archimedean generator means that we can impose certain conditions of regular variation at zero and infinity. This allows for two estimators whose uses are complementary since the conditions required for their asymptotic behavior are different. The case of generators with a finite support is also studied. This chapter establishes consistency and weak convergence of the estimators to a Gaussian process. Chapter 5 consists of an etensive Monte-Carlo type simulation study in which the Mean Integrated Square Error of the two different estimators studied in Chapter 4 is compared. The study includes data simulated from copulas that are known and whose properties are easily tuned by altering the values of their parameters. This allows us to eplore cases in which the estimators are consistent by the results obtained in Chapter 4 as well as cases where the conditions from Chapter 4 do not apply. This leads to interesting comparisons between families and shapes of the Pickands dependence function. Chapter 6 concludes the thesis with a summary of the results and findings in this thesis. Many different paths of research are outlined, resulting from both the assumptions made in Chapter 4 and the simulation study in Chapter 5. 3

17 2 Background Material 2.1 Copulas The joint distribution of several random variables contains the marginal effects as well as the dependence structure which ties them together. While the univariate marginal distributions will describe each individual variable, they do not give any information on how the variables behave together. In general, this behaviour cannot be entirely eplained by a single measure such as linear correlation. Instead, copulas provide all the information about the dependence structure with marginal effects eliminated. In this thesis we will be limiting ourselves to the bivariate case. Note that all the results in this section, including Sklar s theorem, can be etended to an arbitrary number of dimensions. Definition A bivariate copula is the distribution function of a random vector U 1,U 2, where U 1 and U 2 are uniformly distributed on the interval,1. The immediate properties of a bivariate copula C are: 1. C is increasing in each component; 4 2. For all u 1, v 1 [,1], C u 1,= C,u 2 =;

18 2.1. Copulas 3. for all u 1,u 2,u 2,u 2 [,1] such that u 1 u 1 and u 2 u 2, C u 1,u 2 C u 1,u 2 C u 1,u 2 +C u 1,u 2 ; 4. For all u 1,u 2 [,1], C u 1,1=u 1, C 1,u 2 =u 2. Properties 1-3 are due to the fact that C is a bivariate distribution function on the unit square. Property 4 is due to the fact that the marginal distributions are uniform. In fact, these conditions are necessary and sufficient for C to be a bivariate copula. Net, we present a theorem which allows to not only retrieve the copula from a joint distribution but also to build joint distributions using copulas Sklar s Theorem Let X,Y be a pair of random variables with joint distribution function H and marginal distributions F and G. That is, for all, y R, H, y=px,y y, F =PX, Gy=PY y. We can then define the inverses of F and G. For all u 1, v 1,1], F 1 u 1 =inf{ R:F u 1 }, G 1 u 2 =inf{y R: Gy u 2 }. 2.1 These are also known as the quantile functions. Note that they are left-continuous on,1]. The power of copulas for multivariate statistics is mainly due to the following result of Sklar Theorem Given a bivariate distribution function H with marginal distributions F and G, there eists a copula C such that for all, y R, H, y= C {F,Gy}

19 Chapter 2. Background Material Moreover if the marginal distributions F and G are continuous, C is unique and is the cumulative distribution function of the pair U,V = F X,GY. It is given, for all u, v,1], by C u, v= H F 1 u,g 1 v. 2.3 Conversely, given any copula C and univariate distribution functions F and G, the function H defined in 2.2 is a joint distribution function with marginal distributions F and G. In sections pertaining to the stochastic representation of Archimedean and Archima copulas, the notion of survival copula is used. Here we present the survival copula version of Sklar s theorem. Recall that if X,Y are random variables, their bivariate survival function is given by H, y=px >,Y > y. Theorem The following statements are true. 1. If C is a bivariate copula and F and Ḡ are univariate survival functions, then H, y= C F,Ḡy,, y R, 2.4 is a bivariate survival function with marginal survival functions F and Ḡ. 2. Conversely, if H is a bivariate survival function with margins F and Ḡ, there eits a copula C satisfying 2.4. If F and G are continuous, then C is unique and has the form C u, v= H F 1 u,ḡ 1 v for all u, v [,1], where F 1 u=inf { : F u } and Ḡ 1 = inf { y : Ḡy v } Empirical Copulas Let X i,y i n be a random sample from a bivariate distribution function H with continuous margins F and G. First suppose that the margins F and G are known. We then have i=1 access to the copula sample, which is given by 6 U i,v i n i=1 = F X i,gy i n i=1.

20 2.1. Copulas The unknown copula can then be estimated by the empirical distribution function of the sample U i,v i n from C: i=1 C n u, v= 1 n n 1U i u,v i v. i=1 The empirical process α n = C n C has a well known limit. The following theorem can be found in Stute 1984, Gaenssler & Stute 1987, Chapter 5, van der Vaart & Wellner 1996, page 389, Tsukahara 25, Fermanian et al. 24, Ghoudi & Rémillard 24 and van der Vaart & Wellner 27. Remark: Weak convergence is denoted by w throughout this thesis. Theorem The empirical process α n = n 1/2 C n C converges weakly to the bivariate pinned C -Brownian sheet α, i.e. a centered Gaussian random field on the unit square whose covariance is given by cov { αu, v,αu, v } = C u u, v v C u, vc u, v. for all u, v,u, v [,1]. In practice however, the margins are rarely known. When this is the case, a sample from C is no longer observable. To circumvent this problem, we can estimate F and G by the marginal empirical distribution functions. This leads to a pseudo-sample Û i, ˆV i n i=1 from C, where Û i and ˆV i are the scaled component-wise ranks of the observations from H: Û i = 1 n+ 1 n j=1 1 X j X i, ˆV i = 1 n+ 1 n 1 Y j Y i. The unknown copula C can then be estimated from this pseudo-sample. In this thesis, we will be using the same definition of empirical copulas as in Genest & Segers 29. j=1 Definition For all u, v [,1], let the empirical copula be defined as the empirical 7

21 Chapter 2. Background Material distribution function of the scaled component-wise ranks: Ĉ n u, v= 1 n n 1 Û i u, ˆV i v. i=1 As was the case in the situation when F and G are known, it is of interest to study the empirical process n 1/2 Ĉ n C. In order to eploit the result from Theorem 2.1.3, we define the remainder term R n u, v implicitly by the following equation. n 1/2{ Ĉ n u, v C u, v } = α n u, v Ċ 1 u, vα n u,1 Ċ 2 u, vα n 1, v+r n u, v, 2.5 where Ċ 1 and Ċ 2 are the two first order partial derivatives of C, that is C u, v C u, v Ċ 1 u, v=, Ċ 2 u, v=. u v According to Stute 1984 and Tsukahara 25, provided that the second-order partial derivatives of C eist and are continuous on [,1] 2, almost surely as n. sup u,v [,1] 2 R n u, v = O { n 1/4 logn 1/2 loglogn 1/4} 2.6 Note that other definitions of empirical copulas eist, for eample the following definition is based on the fact that C u, v= H F 1 u,g 1 v. Definition Deheuvels 1979 defines the empirical copula as follows. For u, v [,1], Ĉ D n u, v= H n { F n u,g n v}. F n and G n are the empirical distributions of X 1,..., X n and Y 1,...,Y n respectively, while H n is the empirical distribution of the pairs X i,y i n i=1. Equation 2.6 holds true when Ĉ n is replaced by Ĉ D n in Equation 2.5. In fact, when the marginal distributions are continuous, the difference between these two 8

22 2.2. Etreme Value Copulas definitions of the empirical copula is asymptotically negligible. From Equations 2.5 and 2.6, one can obtain the following result. More details can be found in the same reference material given for Theorem 2.1.3, for eample in Tsukahara 25. Theorem The empirical copula processc n = n 1/2 Ĉ n C defined in Equation 2.5 converges weakly to Cu, v=αu, v Ċ 1 u, vαu,1 Ċ 2 u, vα1, v, for all u, v [,1], where α is as defined in Theorem Etreme Value Copulas A particular class of copulas are etreme-value EV copulas, used to model the dependence of multivariate etremes. Again in this section we limit ourselves to the bivariate case. For the case of more dimensions d > 2, see McNeil et al. 25. Consider a random sample X i,y i n from a bivariate distribution H with continuous i=1 margins F and G. We can define the component-wise maima as M n = ma{x 1,..., X n }, N n = ma{y 1,...,Y n }. Their distributions functions are simply given by PM n =F n, PN n y= G n y, 2.7 while their joint distribution function is P M n, N n y = H n, y. First, we can study the univariate case, looking at M n and N n separately. Equations 2.7 are not useful in practice since F and G are unknown. Non-parametric estimates of 9

23 Chapter 2. Background Material F and G are also discarded since raising them to the power of n tends to significantly increase error. The Fisher-Tippett Theorem below allows us to establish the limiting univariate distributions of M n and N n without assuming knowledge of F and G. This result can be seen as an analogue of the central limit theorem where sample means can be approimately described by a normal distribution. Here, the generalized etreme value GEV distribution is used to describe sample maima. Theorem If there eist sequences of constants {a n > } and {a n > } such that P M n b n /a n z Gz as n, for a non-degenerate distribution function G, then G is a member of the GEV family { [ z µ ] } 1/ξ Gz=ep 1+ξ, σ defined on {z : 1+ξz µ/σ>}, where <µ<, σ> and <ξ<. This result is also called the etreme-value theorem. See Embrechts et al. 1997, Theorem 3.2.3, for a sketch of the proof. Note that depending on the shape parameter ξ, the GEV distribution is one of the following three distributions: Fréchet when ξ>, Weibull when ξ < and Gumbel when ξ =. The GEV law is related to the important property of ma-stability, which we define below for univariate distributions. Definition A distribution G is said to be ma-stable if, for every n = 2,3,..., there are constants α n > and β n such that G n α n z+ β n = Gz. The following theorem is essential to the proof of Theorem 2.2.1, since it states that the GEV distribution and univariate ma-stable are in fact the same class of distributions. See, for eample, Coles 21, Theorem 3.2. Theorem A univariate distribution is ma-stable if and only if it is a generalized etreme-value GEV distribution. 1

24 2.2. Etreme Value Copulas We now come back to the bivariate setting. By Sklar s theorem, H the joint distribution function of X and Y has a copula C 1 such that H, y= C 1 F,Gy. Naturally, {F H n, y= C1 n F,G= C 1 n n } 1 n, { G n y } 1 n, and it follows that the unique copula of H n is given by C n u, v= C n 1 u 1 n, v 1 n. 2.8 It can be seen that if C n u, v has a limit for all u, v [,1] as n, this limit, say C, must be an etreme-value copula, as defined below. Definition A bivariate copula is called etreme-value if and only if it has the mastable property for copulas, which states that for any t > and any u, v [,1], C u, v= C t u 1 t, v 1 t. Indeed, suppose that the copula C n given in Equation 2.8 has a limit C u, v as n. Let t N, then Therefore, lim C nt u, v= lim C nt n n 1 u 1 nt, v 1 nt = C u, v. C u, v= lim C nt n 1 = lim n C n 1 u 1 nt, v 1 nt u 1 t 1 n,v 1 t 1 t n = C t u 1 t, v 1 t With appropriate shifting and scaling of M n and N n, these two component-wise maima jointly converge to an etreme-value distribution, say L. That is, the limiting marginal distributions L 1 and L 2 are GEV by Theorem and the underlying limiting copula C is an etreme-value copula as in Definition L is also ma-stable, in the sense that for. 11

25 Chapter 2. Background Material every n there eists constants a n, b n, c n and d n such that for any, y R, L n a n + b n,c n y+ d n =L, y. 2.9 Indeed, fiing n and picking the constants a n, b n, c n and d n such that the ma-stability property from Definition holds for the margins L 1 and L 2 of L, we obtain L n a n + b n,c n y+ d n = C n L 1 a n + b n,l 2 c n y+ d n 2.1 = C n [L n 1 a n+ b n ] 1/n,[L n 2 c n y+ d n ] 1/n = C n [L 1 ] 1/n,[L 2 y] 1/n 2.11 = C L 1,L 2 y = L, y Equation 2.1 is due to Sklar s Theorem. Equation 2.11 follows from the ma-stable property of the marginal distributions L 1 and L 2 which are GEV. Finally, Equations 2.12 are due to the fact that C is a ma-stable copula as well as to Sklar s Theorem Pickands Representation Definition A Pickands dependence function A defined on the unit interval [,1] is a real valued function with the following two properties. 1. A is conve. 2. ma{t,1 t} At 1 for all t [,1]. As shown by Pickands 1981, bivariate etreme-value copulas can be epressed conveniently using a Pickands dependence function A. Theorem A bivariate copula C is an etreme-value copula if and only if there eists a unique Pickands dependence function A such that for u, v,1, [ ] logu C u, v=ep loguva. loguv 12

26 2.2. Etreme Value Copulas The upper bound At = 1 for all t [,1] corresponds to the independence copula C u, v = uv for u, v [,1]. The lower bound At = ma{t,1 t} corresponds to the comonotone copula C u, v = min{u, v} Estimation In this subsection we present two common estimators for the Pickands dependence function. Suppose X,Y is a pair of continuous random variables with a continuous joint distribution function H and marginal distributions F and G. Suppose also the underlying copula C is etreme-value with Pickands dependence function A. Since the variables U = F X and V = GY are then uniformly distributed with copula C, it then follows that S = logu and T = logv are eponentially distributed with mean 1. For t, 1, let ξt= S t T 1 t and set ξ = T and ξ1 = S. The survival function of ξt can be computed, for all t,1, as follows: P ξt> =P S> t,t > 1 t = P U < ep{ t },V < ep{ 1 t} = ep{ At}, where the last step follows from Theorem Therefore ξt is an eponential random variable with mean 1 At. Furthermore, E{logξt}= = logate At d = log y/at e y d y logye y d y logat= γ logat, where γ denotes the Euler-Mascheroni constant. These two observations can now be used to construct non-parametric estimators of the Pickands dependence function A. Suppose that X i,y i n is a random sample from a continuous bivariate distribution H i=1 with an unknown etreme-value copula C. In order to estimate the unknown Pickands de- 13

27 Chapter 2. Background Material pendence function A, recall first that if F and G were known, U i,v i n i=1 = F X i,g Y i n i=1 would constitute a sample from C. Consequently, for any t [,1], ξ i t n i=1 where ξ i 1=logU i, ξ i = logv i, ξ i t= logu i t logv i 1 t, t,1 would constitute a random sample from ξt. The unknown Pickands dependence function A of C could then be estimated by either one of the following statistics: 1/A P n t= 1 n n ξ i t, i=1 log A C n FG t= γ 1 n n logξ i t, for t [,1]. The first estimator is due to Pickands 1981 while the second one, where CFG stands for Capéraà-Fougères-Genest, is due to Capéraà et al The above estimators of A rely on the knowledge of the margins. When F and G are unknown, i.e. when we cannot observe the pairs U i,v i =F X i,g Y i, we can work i=1 with the scaled component-wise ranks instead, viz. Û i = 1 n+ 1 n j=1 1 X j X i, ˆV i = 1 n+ 1 as eplained in Section Now for every i = 1,...,n, let n 1 Y j Y i, Ŝ i = logû i = ˆξ i 1, ˆT i = log ˆV i = ˆξ i, ˆξi t= Ŝ t ˆT 1 t, t,1. The rank-based Pickands and Capéraà-Fougères-Genest estimators as found in Genest & Segers 29 are given by and 1/A P n,r t= 1 n n ˆξ i t, i=1 log A C n,r FG t= γ 1 n j=1 n log ˆξ i t. i=1 14

28 2.3. Archimedean Copulas Among other results, Genest & Segers 29 establish the asymptotic behavior of these two estimators. Define the corresponding empirical processes as follows, A P n,t t=n1/2 A P n,r t At, A C n,t FG t=n 1/2 A C n,r FG t At, for t [,1]. Theorem If A is twice continuously differentiable, thena P n,t w A P r A C FG r FG andac n,t w as n in the space C [,1] of continuous functions on [,1] equipped with the topology of uniform convergence, where 1 A P r t= A2 t C u t,u 1 t du 1 A C r FG t= At u C u t,u 1 t du u logu, wherecdenotes the weak limit of n 1/2 Ĉ n C as in Section In Chapter 4, we will etend these results of Genest & Segers 29 to a broader class of estimators. 2.3 Archimedean Copulas Another widely used copula class is class of Archimedean copulas. A d-variate Archimedean copula has the following form: C ψ u 1,...,u d =ψ ψ 1 u ψ 1 u d, u 1,...,u d [,1] d The right-hand side in Equation 2.13 does not need to be a copula for an arbitrary function ψ. Multivariate Archimedean copulas of dimension d are defined in terms of d-varying Archimedean generators; their eistence has been established in McNeil & Nešlehová 29. We now give the definition of Archimedean generators. Definition Let d 2 be a given integer. The function φ = ψ 1 : [, [,1] is called a d-variate Archimedean generator, if ψ = 1, ψ as and if ψ is 15

29 Chapter 2. Background Material Name C u, v, u, v,1 φ, t [,1] ψ, t, Θ Independence uv logt ep t N/A Clayton u θ + v θ 1/θ 1 t θ θ 1 1+θt 1/θ θ [ 1, \{} [ Gumbel ep logu θ + log v θ] 1/θ log t θ ep t 1/θ θ> 1 uv 1 θ Ali-Mikhail-Haq log 1 θ1 t 1 θ1 u1 v ept θ t θ [ 1,1 Joe 1 1 u θ + 1 v θ 1 u θ 1 v θ 1/θ 1 1 ep t 1/θ log 1 1 t θ θ> 1 e θu 1 e θv 1 Frank 1 θ log 1+ e θ 1 log e θt 1 e θ 1 1+e 1 θ log t e θ 1 Table 2.1: Common Bivariate Archimedean Copulas θ R\{} d-monotone on,, i.e. if ψ has d 2 derivatives satisfying 1 k ψ k on, for 1 k d 2 and that 1 d 2 ψ d 2 is non-increasing and conve on,. By convention, ψ 1 =φ=inf { [, ] : ψ= }. McNeil & Nešlehová 29 prove the following result. Theorem If φ=ψ 1 is a d-variate Archimedean generator, then C u 1,...,u d =ψ ψ 1 u ψ 1 u d, u 1,...,u d [,1] d, is a copula. The Archimedean class encompasses many well-known copula families. Letting the generator be ψt=e t we recover the independence copula C u 1,...,u d =u 1...u d for u 1,...,u d [,1]. Other well-known Archimedean families are the Clayton, Gumbel, Frank and Joe copulas see Table Stochastic Representation In order to obtain a stochastic representation of the Archimedean family, we use the following transformation due to Williamson This representation brings a different perspective on Archimedean copulas and allows us to generate samples using a radial random variable rather than an Archimedean generator. Additionally this stochastic representation is used to construct eamples of generators which are d-monotone. 16

30 2.3. Archimedean Copulas Definition If R is a non-negative random variable with distribution function F R satisfying F R = and d 2 is an integer, then the Williamson d-transform of F R is a real function on [, given by W d F R = 1 d 1 dfr t=e 1 d 1, [,, t R + where EY + denotes the epectation of the non-negative part of the random variable Y. As shown in McNeil & Nešlehová 29, a non-negative random variable is uniquely characterized by its Williamson d-transform for any d 2. In particular, if ψ=w 2 F R, then for,, F R =W 1 2 ψ, where W 1 2 ψ=1 ψ+ ψ In dimensions d > 2, Equation 2.14 generalizes to W 1 d d 2 ψ=1 k= 1 k k ψ k k! 1d 1 d 1 ψ d 1 +. d 1! McNeil & Nešlehová 29 link simple distributions to Archimedean copulas via the Williamson d-transform. In the bivariate case, a random vector X=X 1, X 2 is said to have a simple distribution if X= d RS 2 where S 2 is a random vector uniformly distributed on the unit simple S 2 = { R 2 + : 1= 1 }, and R is an independent, strictly positive scalar random variable. The radial distribution refers to F R, the distribution function of the independent strictly positive scalar random variable R.. 1 is the l 1 norm, i.e. for = 1,...,d d, 1 = d i. i=1 Theorem The following statements hold: If X = X 1, X 2 has a simple distribution with radial distribution F R satisfying F R =, then X has a bivariate Archimedean survival copula with generator ψ= W 2 F R. 17

31 Chapter 2. Background Material If U,V is distributed as a bivariate Archimedean copula C with generator ψ, then ψ 1 U,ψ 1 V has a simple distribution with radial distribution F R =W 1 2 ψ Eamples McNeil & Nešlehová 21 use the characterization in Theorem to create new copula families, including the following. The Pareto-Simple Family Suppose that R is Pareto with distribution function F R r =1 r α for r 1 and F R r = for r < 1 and parameter α>. As shown in McNeil & Nešlehová 21, the Archimedean copula of the d-dimensional simple distribution has the following generator: ψ α,d =α α B min,1,α,d where B denotes the incomplete beta function. In two dimensions the generator simply reduces to 1 α α+1 for 1 ψ α,2 = α for > 1 α+1. The Inverse Pareto-Simple Family If 1/R is Pareto with parameter γ>, then the density of R is given by f r r =γr γ 1 for r,1] and f r r = otherwise. In two dimensions, the resulting generator can easily be found. For [,1], ψ γ,2 = 1 1 r γr γ 1 dr = γ 1 r γ 1 dr 1 r γ 2 dr = 1 γ γ γ 1 + γ γ

32 2.3. Archimedean Copulas = 1 γ γ 1 + γ γ 1, 2.15 and ψ γ,2 = for > 1. The Miture Family Let R have distribution function for < F R = γ for 1, 2 1 α 2 for > 1 meaning R is drawn with probability 1/2 from a Pareto distribution with parameter α and with probability 1/2 from an inverse Pareto distribution with parameter γ. In two dimensions, the generator is given by the net epression. ψ α,γ,2 = 1 γ 2 γ γ 2 γ 1 α The derivations are shown below. First, for > 1, ψ α,γ,2 =W 2 F R = Now if 1 we obtain ψ α,γ,2 =W 2 F R = r 1 for 1 2α α α+1 for > 1 αr α 1 dr = α r α 1 dr r α 2 dr 2 2 = α [ α α 1]= 2 α α α+1 2α+1. 1 γ r 2 r α 1 dr + = γ 2 1 r γ 1 dr r r γ 2 dr + α 2 αr α 1 dr 2 [ 1 α ] α+1 19

33 Chapter 2. Background Material { 1 = 2 2 γ γ γ = 1 γ 2 γ γ 2 } { γ 1 γ α } 2α+1 γ 1 α 2α Random Number Generation The stochastic representation of Archimedean copulas from Theorem makes it easy to generate random samples. To generate a sample from a bivariate Archimedean copula C ψ with generator ψ, proceed as following. 1. Generate R, usually by using inverse probability sampling. If we start from the generator ψ, we can use Equation Independently, generate a random vector S 2 uniformly distributed on the unit simple S 2 by using Y 1 Y 2 S 2 = S 1,S 2 = d, Y 1 + Y 2 Y 1 + Y 1 where Y 1,Y 2 are independent eponential random variables with unit mean. 3. The observation from C ψ is given by ψrs1,ψrs 2. This algorithm was first introduced by Whelan 24, but only justified for completely monotone generators. McNeil & Nešlehová 29 show that this procedure can be used for any d-monotone generator. 2.4 Regular Variation Regular variation is a concept crucial to the estimators used in this thesis. Regular variation of the Archimedean generator was found to be one of the sufficient conditions to 2

34 2.4. Regular Variation ensure the consistency of the proposed estimators. Definition A measurable function f :,, is called slowly varying at infinity if for all t >, f t lim f = 1. Slowly varying functions include any power of the logarithmic function, as well as any function with a positive limit at infinity. Definition A measurable function f :,, is called regularly varying function at infinity with inde α R written f RV α, if for any >, f t lim t f = α. R denotes the etended real number line. The following result is often used in Chapter 4. For a proof, see Theorem.8v from Resnick Theorem Suppose that f is non decreasing, f =, and f RV ρ, ρ. Then f 1 RV ρ 1. where f 1 y=inf { : f y }, as in Equation 2.1. Another important result is Karamata s theorem, see Theorem.6 in Resnick Theorem The following statements hold. 1. If ρ 1 then f RV ρ implies f td t RV ρ+1 and If ρ< 1 or if ρ= 1 and f RV ρ+1 and lim f = ρ+ 1. f td t f sd s< then f RV ρ implies lim f f td t = ρ 1. f td t is finite, 21

35 Chapter 2. Background Material 2. If f satisfies then f RV λ 1. If lim f td t < and lim f = λ,, f td t f f td t = λ,, then f RV λ 1. We finish this section with another important result called Karamata s characterization theorem. Theorem If f :,, is regularly varying at infinity with inde ρ, then there eists a slowly varying function L such that f = ρ L. Proof. Let L = f / p. This function has inde of regular variation equal to zero, meaning that it is slowly varying. Combining the previous two theorems, we have that if f RV ρ with ρ < 1, then for a suitable slowly varying function L, f td t = ρ+1 L, 2.17 which will prove to be quite useful in Chapter 4. 22

36 3 Archima Copulas In the previous chapter, definitions of the Pickands dependence function and the Archimedean generator were given. These two functions are used jointly in the definition of a bivariate Archima copula. Definition A bivariate Archima copula is given by the following epression, valid for all u 1,u 2 [,1]: [ C ψ,a u 1,u 2 =ψ {ψ 1 u 1 +ψ 1 ψ 1 ] u 1 u 2 }A ψ 1 u 1 +ψ 1, u 2 where A : [,1] [1/2,1] is a Pickands dependence function and φ = ψ 1 a 2-variate Archimedean generator. The proof that C ψ,a is indeed a copula can be found in Appendi A of Capéraà et al. 2. This family of copulas is a generalization of both Archimedean copulas and etreme value copulas. Indeed, using the generator φ EV t=ψ 1 EV t= logt, t [,, 23

37 Chapter 3. Archima Copulas we obtain the etreme-value copula with Pickands dependence function A: logu1 C ψev,au 1,u 2 =ep{ +logu 2 logu 1 A logu 1 +logu 2 }=u 1 u 2 A logu 1 logu 1 +logu 2 When using the identity function A I = 1, we recover the Archimedean copula with generator ψ: C ψ,ai u 1,u 2 =ψ [ {ψ 1 u 1 +ψ 1 u 2 } ]. Capéraà et al. 2 establish the maimum domain of attraction MDA of certain bivariate Archima copulas in the net theorem. That is, they determine the limit C of. C n u, v= C n ψ,a u1/n, v 1/n, u, v [,1] as n. As seen in Section 2.2, if the limit eists, then C is an etreme-value copula and C is said to be in the maimum domain of attraction of C. Theorem If C ψ,a is a bivariate Archima copula with Pickands dependence function A and Archimedean generator φ=ψ 1 such that φ1 1/t RV m for some m 1, then C ψ,a belongs to the maimum domain of attraction of an etreme value copula with Pickands dependence function A given by A t= { t m + 1 t m} A 1/m { t m t m + 1 t m }, t [,1]. Analogously, Capéraà et al. 2 also establish the minimum domain of attraction of certain bivariate Archima copulas. This is done by computing the limit of C n u, v= C n ψ,a 1 u 1/n,1 v 1/n, as n, where C denotes a survival copula. Theorem If C ψ,a is a bivariate Archima copula with Pickands dependence function A and Archimedean generator φ=ψ 1 such that lim t φt= and φ RV 1/m for some m, then C ψ,a belongs to the minimum domain of attraction of the copula C u, v= u+ v 1+1 u1 v/c ψ,a1 u,1 v, where ψ 1 t=log 1/m 1/t. 24

38 3.1. Stochastic representation Note that C ψ,a1 u,1 v in Theorem 3..5 is not a copula because ψ 1 t=log 1/m 1/t is not a 2-variate Archimedean generator. Theorems 3..4 and 3..5 are generalized to d > 2 dimensions in Propositions 5.1 and 5.4 respectively of Charpentier et al They are important in that they motivate the construction of Archima copulas. Indeed, they allows us to construct parametric families of bivariate copulas with given limiting etreme-value attractors. As such, Archima copulas provide etra fleibility when compared to etreme-value copulas. They can be used in pre-asymptotic settings, i.e., when the data at hand do not yet ehibit the limiting etreme-value dependence. 3.1 Stochastic representation In analogy to Archimedean copulas in Subsection 2.3.1, we present a stochastic representation of Archima copulas. For the sake of generality, we will present results for any number of dimensions first and focus on the special case of two dimensions subsequently. To do so, a new type of function is called upon to characterize Archima copulas. This following function is defined in Huang Definition A function is l : [, [,1] is called a d-variate stable tail dependence function if there eists a d-variate etreme-value copula D such that for all 1,..., d [,, l 1,..., d = log { D e 1,...,e } d. In the bivariate case, the stable tail dependence function is linked to the Pickands dependence function by the following equation, valid for all, y [,, l, y=+ya +y Charpentier et al. 214 prove that for any choice of d-variate stable tail dependence function l and d-variate Archimedean generator ψ,. C ψ,l u 1,...,u d =ψ l { φu 1,...,φu d } 25

39 Chapter 3. Archima Copulas is an Archima copula. Charpentier et al. 214 then give two stochastic representations for Archima copulas. The first relies on a random vector T 1,...,T d with survival function given for all t 1,..., t d [, by PT 1 > t 1,...,T 2 > t d =ep{ lt 1,..., t d }. 3.1 Each T i for i {1,...,d} is eponentially distributed with unit mean, while the survival copula of the random vector T 1,...,T d is an etreme-value copula with stable tail dependence function l. In the bivariate case, the joint survival function of T 1,T 2 is then t 1 PT 1 > t 1,T 2 > t 2 =ep t 1 + t 2 A, t 1, t 2. t 1 + t 2 Theorem The copula C ψ,l is Archima with d-variate stable tail dependence function l and completely monotone Archimedean generator ψ if and only if it is the survival copula of the random vector X 1,..., X d =T 1 /Θ...,T d /Θ, where Θ has Laplace transform ψ and is stochastically independent of the random vector T 1,...,T d defined in Equation 3.1. This representation only holds for completely monotone generators ψ. As we saw in Theorem 2.3.1, we require only d-monotonicity. For this more general case, consider a random vector S 1,...,S d of strictly positive random variables such that for all s 1,..., s d [,, PS 1 > s 1,...,S d > s d = Ḡ l s 1,..., s d =[ma{,1 ls 1,..., s d }] d Note that in the bivariate case, 26 [ { }] s 1 PS 1 > s 1,S 2 > s 2 = Ḡ l s 1, s 2 = ma,1 s 1 + s 2 A, s 1, s 2. s 1 + s 2