Uniform consistency of local likelihood estimators for covariate-dependent copula parameters

Abstract

In many applications, the dependence structure among multiple random variables varies with observed covariates. Conditional copula models provide a flexible framework for representing such covariate-driven dependence while preserving a separation between marginal behavior and association. This paper studies the uniform asymptotic behaviour of local likelihood estimators for smoothly varying copula parameters in multivariate conditional copula models. Assuming that the conditional dependence structure is governed by a parametric copula whose parameter is a smooth function of a covariate vector, we estimate the associated calibration function using a kernel-weighted local likelihood approach combined with a suitable link transformation. Within a multivariate local polynomial framework, we establish uniform convergence rates over compact covariate sets for the local log-likelihood, its score, and its Hessian. These results imply uniform consistency of the local maximum likelihood estimator of the calibration function and of the induced copuThere exist measurable envelope functionsla parameter function. The analysis relies on empirical process techniques for kernel-indexed classes with shrinking neighbourhoods and polynomial entropy bounds, allowing control of stochastic fluctuations uniformly over both covariate locations and local polynomial coefficients. The resulting theory provides a rigorous foundation for global consistency, numerical stability of local optimization algorithms, and the theoretical justification of data-driven bandwidth selection in covariate-dependent copula models.