Information bounds for Gaussian copulas

TL;DR

This paper derives the asymptotic distribution of the rank likelihood in Gaussian copula models, showing that rank-based estimators achieve the same information bounds as full-data estimators, with multivariate normal distributions being least favorable.

Contribution

It provides the limiting normal distribution of the rank likelihood for Gaussian copula models, including structured correlation matrices, and establishes the equivalence of semiparametric and parametric information bounds.

Findings

Limiting distribution of rank likelihood ratio matches that of multivariate normal models.

Semiparametric information bounds are equal to full-data bounds for Gaussian copulas.

Multivariate normal distributions are least favorable in this context.

Abstract

Often of primary interest in the analysis of multivariate data are the copula parameters describing the dependence among the variables, rather than the univariate marginal distributions. Since the ranks of a multivariate dataset are invariant to changes in the univariate marginal distributions, rank-based estimators are natural candidates for semiparametric copula estimation. Asymptotic information bounds for such estimators can be obtained from an asymptotic analysis of the rank likelihood, that is, the probability of the multivariate ranks. In this article, we obtain limiting normal distributions of the rank likelihood for Gaussian copula models. Our results cover models with structured correlation matrices, such as exchangeable or circular correlation models, as well as unstructured correlation matrices. For all Gaussian copula models, the limiting distribution of the rank likelihood ratio is shown to be equal to that of a parametric likelihood ratio for an appropriately chosen multivariate normal model. This implies that the semiparametric information bounds for rank-based estimators are the same as the information bounds for estimators based on the full data, and that the multivariate normal distributions are least favorable.