Uniform Asymptotic Theory for Local Likelihood Estimation of Covariate-Dependent Copula Parameters

TL;DR

This paper develops a comprehensive asymptotic theory for local likelihood estimation of covariate-dependent copula parameters, ensuring uniform convergence and consistency in multivariate models with varying dependence structures.

Contribution

It introduces a uniform asymptotic framework for kernel-weighted local likelihood estimators in covariate-dependent copula models, with theoretical guarantees.

Findings

Establishes uniform convergence rates for local likelihood estimators.

Proves uniform consistency of the copula parameter function.

Provides empirical process techniques for kernel-indexed classes.

Abstract

Conditional copula models allow dependence structures to vary with observed covariates while preserving a separation between marginal behavior and association. We study the uniform asymptotic behavior of kernel-weighted local likelihood estimators for smoothly varying copula parameters in multivariate conditional copula models. Using a local polynomial approximation of a suitably transformed calibration function, we establish uniform convergence rates over compact covariate sets for the local log-likelihood, its score, and its Hessian. These results yield uniform consistency of the local maximum likelihood estimator and of the induced copula parameter function. The analysis is based on empirical process techniques for kernel-indexed classes with shrinking neighborhoods and polynomial entropy bounds, providing theoretical support for global consistency and stable local optimization in covariate-dependent copula models.