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New copula families and mixing properties

Martial Longla

Abstract

We characterize absolutely continuous symmetric copulas with square integrable densities in this paper. This characterization is used to create new copula families, that are perturbations of the independence copula. The full study of mixing properties of Markov chains generated by these copula families is conducted. An extension that includes the Farlie-Gumbel-Morgenstern family of copulas is proposed. We propose some examples of copulas that generate non-mixing Markov chains, but whose convex combinations generate $ψ$-mixing Markov chains. Some general results on $ψ$-mixing are given. The Spearman's correlation $ρ_S$ and Kendall's $τ$ are provided for the created copula families. Some general remarks are provided for $ρ_S$ and $τ$. A central limit theorem is provided for parameter estimators in one example. A simulation study is conducted to support derived asymptotic distributions for some examples.

New copula families and mixing properties

Abstract

We characterize absolutely continuous symmetric copulas with square integrable densities in this paper. This characterization is used to create new copula families, that are perturbations of the independence copula. The full study of mixing properties of Markov chains generated by these copula families is conducted. An extension that includes the Farlie-Gumbel-Morgenstern family of copulas is proposed. We propose some examples of copulas that generate non-mixing Markov chains, but whose convex combinations generate -mixing Markov chains. Some general results on -mixing are given. The Spearman's correlation and Kendall's are provided for the created copula families. Some general remarks are provided for and . A central limit theorem is provided for parameter estimators in one example. A simulation study is conducted to support derived asymptotic distributions for some examples.
Paper Structure (16 sections, 15 theorems, 93 equations, 4 figures)

This paper contains 16 sections, 15 theorems, 93 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Square integrable symmetric densities
  3. Perturbations of the copula $\Pi(u,v)$
  4. Examples of new copula families
  5. Copulas based trigonometric on functions
  6. The extended Farlie - Gumbel - Morgenstern copulas
  7. Counter example - the sine basis
  8. Copula-based Markov chains and $\psi$-mixing
  9. Applications to our new copula families
  10. Mixing properties of the extended FGM family
  11. Answer to an open question on mixing
  12. Central limit theorem and simulation study
  13. Central limit theorem for parameter estimators
  14. Simulation study
  15. Conclusion and comments
  16. ...and 1 more sections

Key Result

Proposition 2.1

For any absolutely continuous copula $C(u,v)$ the following holds.

Figures (4)

  • Figure 1: Examples of sine-cosine copula densities
  • Figure 2: Examples of sine copula densities
  • Figure 3: Densities of Legendre-Farlie-Gumbel-Morgenstern copulas
  • Figure 4: Density of copula $C_{1,\lambda}$

Theorems & Definitions (24)

  • Proposition 2.1
  • Remark 1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Remark 3
  • Lemma 3
  • Remark 4
  • ...and 14 more