Minimax Lower Bounds for Uniform Estimation of Covariate-Dependent Copula Parameters

Local likelihood methods are widely used to estimate calibration functions in conditional copula models. Recent work has established uniform stochastic equicontinuity and uniform convergence rates for local likelihood estimators of covariate-dependent copula parameters, yielding global consistency guarantees and supporting the stability of local optimization routines. This paper complements those results by deriving minimax lower bounds for uniform estimation over Hölder classes of calibration functions. Under mild regularity conditions on the copula family and the covariate design, we show that the minimax sup-norm risk over a compact covariate region is bounded below by the classical nonparametric rate for smooth functions on an s-dimensional domain. The proof combines a localized packing construction with a Fano–Le Cam testing argument, using second-order expansions of the conditional copula likelihood to control information distances. As a consequence, local polynomial likelihood estimators achieve the minimax rate up to the logarithmic factors inherent to uniform estimation, providing a sharp optimality justification for their use in conditional copula modeling.