Information bounds for Gaussian copulas

Area of Science:

• Statistics
• Multivariate Data Analysis
• Copula Theory

Background:

• Copula parameters are crucial for understanding multivariate data dependence.
• Rank-based estimators offer a semiparametric approach to copula estimation, invariant to marginal distributions.
• Analyzing the rank likelihood is key to determining asymptotic information bounds.

Purpose of the Study:

• To derive the limiting normal distributions of the rank likelihood for Gaussian copula models.
• To investigate these distributions for both structured and unstructured correlation matrices.
• To compare the asymptotic properties of rank-based estimators with parametric approaches.

Main Methods:

• Asymptotic analysis of the rank likelihood function.
• Derivation of limiting normal distributions for rank likelihood ratios.
• Consideration of Gaussian copula models with various correlation structures.

Main Results:

• The limiting distribution of the rank likelihood ratio for Gaussian copulas matches that of a parametric likelihood ratio.
• This holds true for both structured (e.g., exchangeable, circular) and unstructured correlation matrices.
• Semiparametric information bounds for rank-based estimators are equivalent to parametric bounds.

Conclusions:

• Rank-based semiparametric copula estimation achieves the same information bounds as parametric methods.
• Multivariate normal distributions are identified as least favorable in this context.
• The findings support the use of rank-based methods for robust dependence analysis in Gaussian copulas.