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Clustered Archimax Copulas

Simon Chatelain, Samuel Perreault, Johanna G. Nešlehová, Anne-Laure Fougères

TL;DR

The class of Archimax copulas is extended via their stochastic representation to a clustered construction, which allows for both asymptotic dependence and indepen- dence between the clusters, a property sought in applications in environmental sciences andnance.

Abstract

When modeling multivariate phenomena, properly capturing the joint extremal behavior is often one of the many concerns. Archimax copulas appear as successful candidates in case of asymptotic dependence. In this paper, the class of Archimax copulas is extended via their stochastic representation to a clustered construction. These clustered Archimax copulas are characterized by a partition of the random variables into groups linked by a radial copula; each cluster is Archimax and therefore defined by its own Archimedean generator and stable tail dependence function. The proposed extension allows for both asymptotic dependence and independence between the clusters, a property which is sought, for example, in applications in environmental sciences and finance. The model also inherits from the ability of Archimax copulas to capture dependence between variables at pre-extreme levels. The asymptotic behavior of the model is established, leading to a rich class of stable tail dependence functions.

Clustered Archimax Copulas

TL;DR

The class of Archimax copulas is extended via their stochastic representation to a clustered construction, which allows for both asymptotic dependence and indepen- dence between the clusters, a property sought in applications in environmental sciences andnance.

Abstract

When modeling multivariate phenomena, properly capturing the joint extremal behavior is often one of the many concerns. Archimax copulas appear as successful candidates in case of asymptotic dependence. In this paper, the class of Archimax copulas is extended via their stochastic representation to a clustered construction. These clustered Archimax copulas are characterized by a partition of the random variables into groups linked by a radial copula; each cluster is Archimax and therefore defined by its own Archimedean generator and stable tail dependence function. The proposed extension allows for both asymptotic dependence and independence between the clusters, a property which is sought, for example, in applications in environmental sciences and finance. The model also inherits from the ability of Archimax copulas to capture dependence between variables at pre-extreme levels. The asymptotic behavior of the model is established, leading to a rich class of stable tail dependence functions.
Paper Structure (23 sections, 8 theorems, 99 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 8 theorems, 99 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model specification and notation
  4. Extremal properties
  5. Insights into modeling
  6. Inference when $\mathcal{G}$ is known
  7. Individual cluster inference
  8. Estimating $\psi_1,\dots,\psi_{K}$
  9. Estimating $\ell_1,\dots,\ell_K$
  10. Inference for the dependence between distortions
  11. Data illustration
  12. Data preprocessing
  13. Inference for $\bm{\psi}$ and $\bm{\ell}$
  14. Inference for $Q$
  15. Discussion
  16. ...and 8 more sections

Key Result

Theorem 4.1

Let $C_{\mathcal{G},\bm{\psi},\bm{\ell}, Q}$ be a clustered Archimax copula such that Assumptions assumption:MDA and assumption:stdf hold with $(W_1,\ldots, W_K)$ independent of $\bm{S}$. For $k\in\mathcal{D}_1$, let $b_k= \mathop{\mathrm{E}}\nolimits\{1/Z_k^{\rho_k} \}$, $Z_k\sim \mathrm{Beta}(1,d_

Figures (10)

  • Figure 5.1: Pair chi plots for the sample from Model A (left) and Model B (right). The full black line is the empirical estimate of \ref{['eq:chi']}, the dotted lines are 95% confidence intervals and the red lines represent the true values of $\lim_{q\to 1}\chi_{ij}(q)$ for each pair ${(i,j)}$. The dotted green line in panel (f) represents the upper tail dependence coefficient of $(1/R_2,1/R_3)$. The samples used for the empirical estimates are those of Figures \ref{['fig:pairplotN']} and \ref{['fig:pairplotG']}
  • Figure 7.1: Left: Geographical position of the stations used in the application of Section \ref{['sec:data-application']}, along with their respective cluster label. Right: Pair plots of the scaled componentwise ranks of monthly maxima of precipitations for the stations around which the clusters were formed.
  • Figure 7.2: Estimated quantities from Section \ref{['sec:app-inference']}. Left: Matrix of intra-cluster pairwise upper tail dependence coefficients ($d=22$ stations). Right: Matrix of pairwise distortion parameters ($d=23$ stations).
  • Figure D.1: (Model A) Pairwise plots of a sample of size $n=1000$ from the copula $C_{\mathcal{G},\bm{\psi},\bm{\ell},Q}$ where $\mathcal{G}=\{\mathcal{G}_1,\mathcal{G}_2,\mathcal{G}_3 \}=\{\{1,2,3\},\{4,5,6\},\{7,8,9\} \}$. $\bm{\psi}=\{\psi_{\theta_1},\psi_{\theta_2},\psi_{\theta_3} \}$ with $(\theta_1,\theta_2,\theta_3)=(1.5,1.5,2)$ where $\psi_{\theta_1}$ is Clayton while $\psi_{\theta_2},\psi_{\theta_3}$ are Joe. $\bm{\ell}=\{\ell_{\vartheta_1},\ell_{\vartheta_2},\ell_{\vartheta_3} \}$ with $(\vartheta_1,\vartheta_2,\vartheta_3)=(1.25,2,1.5)$ where $\ell_{\vartheta_1},\ell_{\vartheta_2},\ell_{\vartheta_3}$ are Gumbel-Hougaard. The radial survival copula $\bar{D}$ is trivariate Gaussian with correlations all equal to $0.5$. Upper: linear correlation, Lower: contour density, Diagonal: univariate histogram.
  • Figure D.2: (Model B) Pair plots of a sample of size $n=1000$ from the copula $C_{\mathcal{G},\bm{\psi},\bm{\ell},Q}$ where $\mathcal{G}=\{\mathcal{G}_1,\mathcal{G}_2,\mathcal{G}_3 \}=\{\{1,2,3\},\{4,5,6\},\{7,8,9\} \}$. $\bm{\psi}=\{\psi_{\theta_1},\psi_{\theta_2},\psi_{\theta_3} \}$ with $(\theta_1,\theta_2,\theta_3)=(1.5,1.5,2)$ where $\psi_{\theta_1}$ is Clayton while $\psi_{\theta_2},\psi_{\theta_3}$ are Joe. $\bm{\ell}=\{\ell_{\vartheta_1},\ell_{\vartheta_2},\ell_{\vartheta_3} \}$ with $(\vartheta_1,\vartheta_2,\vartheta_3)=(1.25,2,1.5)$ where $\ell_{\vartheta_1},\ell_{\vartheta_2},\ell_{\vartheta_3}$ are Gumbel-Hougaard. The radial survival copula $\bar{D}$ is trivariate Gumbel-Hougaard with parameter $\vartheta_R=4$. Upper: linear correlation, Lower: contour density, Diagonal: univariate histogram.
  • ...and 5 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 3.1
  • Theorem 4.1
  • Corollary 4.1
  • Remark 4.1
  • Corollary 4.2
  • Remark 4.2
  • Example 5.1: Clayton Generator
  • Example 5.2: Joe generator
  • Remark 5.1
  • ...and 16 more