Empirical likelihood based confidence regions for functional of copulas

Abstract

In the present paper, we are mainly concerned with the statistical inference for the functional of nonparametric copula models satisfying linear constraints. The asymptotic properties of the obtained estimates and test statistics are given. Finally, a general notion of bootstrap for the proposed estimates and test statistics, constructed by exchangeably weighting sample, is presented, which is of its own interest. These results are proved under some standard structural conditions on some classes of functions and some mild conditions on the model, without assuming anything about the marginal distribution functions, except continuity. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.

Keywords:

• Dependence function
• semiparametric statistics
• empirical processes
• weak convergence
• donsker classes
• empirical likelihood
• general bootstrap
• exchangeability

Mathematics Subject Classifications:

• 62F03
• 62F10
• 62F12
• 62H12
• 62H15

Acknowledgments

The authors thank the Associate Editor and two anonymous referees for their thoughtful comments and suggestions which substantially helped to improve the presentation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It seems that Höffding (1940) also had the basic idea of summarising the dependence properties of a multivariate distribution by its corresponding copula, but he chose to define the corresponding function on $[-\frac{1}{2},\frac{1}{2}{]}^d$ rather than on $[0,1{]}^d$. In particular, see the translation of the Höffding (1940) paper in Hoeffding (1994).

2 Condition (6(6) $0<\underset{\mathbf{u}\in [0,1{]}^d}{inf}\frac{d{\mathbb{Q}}_0}{d{\mathbb{C}}_T}(\mathbf{u})\le \underset{\mathbf{u}\in [0,1{]}^d}{sup}\frac{d{\mathbb{Q}}_0}{d{\mathbb{C}}_T}(\mathbf{u})<+\mathrm{\infty },{\mathbb{P}}_T-a.s.$(6) ) means that ${\mathbb{P}}_T(\mathbf{u}\in [0,1{]}^d:\frac{{\displaystyle \mathrm{d}{\mathbb{Q}}_0}}{{\displaystyle \mathrm{d}{\mathbb{C}}_T}}(\mathbf{u})\le 0)={\mathbb{P}}_T(\mathbf{u}\in [0,1{]}^d:\frac{{\displaystyle \mathrm{d}{\mathbb{Q}}_0}}{{\displaystyle \mathrm{d}{\mathbb{C}}_T}}(\mathbf{u})=+\mathrm{\infty })=0.$