Bernstein Empirical Copula Smoothing for Rank-Based Dependence Estimation

Accepted July 2026

Abstract

This paper proposes a Bernstein empirical copula estimator of Kendall’s tau. The estimator remains rank-based and replaces the empirical copula in the Kendall functional by its Bernstein-smoothed version, thereby introducing a tunable bias–variance trade-off. Its bias, variance, mean squared error and asymptotic normality are studied using Bernstein empirical copula process arguments. An explicit functional variance-improvement coefficient is defined, and the independence case is examined separately. Under exact independence and for fixed Bernstein order, the proposed estimator has a smaller leading MSE than the classical Kendall
estimator. Monte Carlo results show that Bernstein smoothing improves MSE mainly under weak and mild dependence, while smoothing bias dominates under stronger dependence. Thus, its main practical advantage is concentrated in weakly dependent settings. A real-data example based on the USArrests data uses a study-specific repeated five-fold MSE-oriented cross-validation criterion to select the Bernstein order.