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On differentiability and mass distributions of topologically typical multivariate Archimedean copulas

Nicolas Dietrich, Wolfgang Trutschnig

Abstract

Copulas, in particular Archimedean copulas are commonly viewed as analytically nice and regular objects. Motivated by a recently established result sta\-ting that the first partial derivatives of bivariate copulas can exhibit surprisingly pathological behavior, we focus on the class of $d$-dimensional Archimedean copulas denoted by $\mathcal{C}_{ar}^d$ and show that partial derivatives of order $(d-1)$ can be sur\-pri\-singly irregular as well. In fact, we prove the existence of Archimedean copulas $C \in \mathcal{C}_{ar}^d$ whose $(d-1)$-st order partial derivatives are pathological in the sense that for almost every $\mathbf{x} \in [0,1]^{d-1}$ the derivative $\partial_1...\partial_{d-1}C(\mathbf{x},y)$ does not exist on a dense set of $y \in (0,1)$. \\ Since the existence of mixed partial derivatives of order $(d-1)$ of a copula $C$ is closely related to the existence of a discrete component, we also study mass distributions of Archimedean copulas. Building upon the interplay between Archimedean copulas and so-called Williamson measures we show that absolute continuity, discreteness and singularity of the Williamson measure propagates to the associated Archimedean copula and vice versa. Moreover, we prove the fact that the sub-family of $\mathcal{C}_{ar}^d$ consisting of copulas whose absolutely continuous, discrete and singular component have full support is dense in $\mathcal{C}_{ar}^d$. \\ Finally, viewing $\mathcal{C}_{ar}^d$ in the light of Baire categories, we show that, in contrast to the space of bivariate copulas, a topologically typical $d$-dimensional Archimedean copula $C$ is not absolutely continuous but has degenerated discrete component, implying that pathological elements are rare in $\mathcal{C}_{ar}^d$.

On differentiability and mass distributions of topologically typical multivariate Archimedean copulas

Abstract

Copulas, in particular Archimedean copulas are commonly viewed as analytically nice and regular objects. Motivated by a recently established result sta\-ting that the first partial derivatives of bivariate copulas can exhibit surprisingly pathological behavior, we focus on the class of -dimensional Archimedean copulas denoted by and show that partial derivatives of order can be sur\-pri\-singly irregular as well. In fact, we prove the existence of Archimedean copulas whose -st order partial derivatives are pathological in the sense that for almost every the derivative does not exist on a dense set of . \\ Since the existence of mixed partial derivatives of order of a copula is closely related to the existence of a discrete component, we also study mass distributions of Archimedean copulas. Building upon the interplay between Archimedean copulas and so-called Williamson measures we show that absolute continuity, discreteness and singularity of the Williamson measure propagates to the associated Archimedean copula and vice versa. Moreover, we prove the fact that the sub-family of consisting of copulas whose absolutely continuous, discrete and singular component have full support is dense in . \\ Finally, viewing in the light of Baire categories, we show that, in contrast to the space of bivariate copulas, a topologically typical -dimensional Archimedean copula is not absolutely continuous but has degenerated discrete component, implying that pathological elements are rare in .

Paper Structure

This paper contains 9 sections, 35 theorems, 96 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. General notation and definitions
  4. Multivariate Archimedean copulas
  5. Regularity and mass distributions of multivariate Archimedean copulas
  6. Regularity of the Kendall distribution function of Archimedean copulas
  7. Derivatives of multivariate Archimedean copulas
  8. Baire category results for multivariate Archimedean copulas
  9. Denseness of Williamson measures with full support

Key Result

Lemma 2.1

The maps $\mathcal{W}_d$, $\xi_d$ and $\xi_d \circ \mathcal{W}_d$ are homeomorphisms.

Figures (5)

  • Figure 1: The distribution function $F_\gamma$ (left panel) and the associated generator $\psi$ (right panel) considered in Example \ref{['ex:supp_arch_luecken']}.
  • Figure 2: Sample of size 10000 of the Archimedean copula $C_\psi$ with $\psi$ being the generator from Example \ref{['ex:supp_arch_luecken']}, its histogram and the two marginal histograms; sample generated via conditional inverse sampling.
  • Figure 3: Plots of the distribution function $F_\gamma$ (left panel) of the Williamson measure $\gamma$ considered in Example \ref{['ex:non_diff']}, the associated generator $\psi$ (middle) and the graphs of the functions $f^a$ with $a \in \{0,\frac{225}{392}\}$ of the corresponding Archimedean copula $C$ (right panel).
  • Figure 4: The distribution function $F_\gamma$ (left panel) and the associated generator $\psi$ (right panel) considered in Example \ref{['ex:arch_abs_and_ex']}.
  • Figure 5: Sample of size 10000 of the Archimedean copula $C_\psi$ with $\psi$ being the generator from Example \ref{['ex:arch_abs_and_ex']}, its histogram and the two marginal histograms; sample generated via conditional inverse sampling.

Theorems & Definitions (68)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Example 3.5
  • ...and 58 more