Asymptotic Theory of Tail Dependence Measures for Checkerboard Copula and the Validity of Multiplier Bootstrap

Abstract

In this paper, we develop a comprehensive asymptotic and bootstrap theory for checkerboard-based estimation of lower and upper tail copulas under unknown marginal distributions. The estimator is constructed via local bilinear (checkerboard) interpolation of the empirical copula and extended to the tail region to obtain nonparametric estimators of extremal dependence. We first establish almost sure uniform consistency of the checkerboard-smoothed copula estimator by decomposing the error into a stochastic empirical process term and a deterministic approximation bias induced by the checkerboard projection. Under mild growth conditions on the grid size, the estimator is shown to be strongly consistent. Next, we derive weak convergence of the centered and scaled checkerboard copula process in $\ell^\infty([0,1]^2)$, showing that the smoothing does not affect the first-order limit. The resulting Gaussian process coincides with that of the empirical copula, augmented by terms arising from marginal estimation. These results extend to the lower and upper tail copula processes, yielding functional central limit theorems and asymptotic normality of the tail dependence coefficient. Since the limiting covariance depends on unknown tail features and partial derivatives rendering direct inference infeasible, we propose a direct multiplier bootstrap adapted to the checkerboard structure. We prove conditional weak convergence of the bootstrap process to the same limit, ensuring valid inference for smooth functionals. Finally, we illustrate the bootstrap methodology through simulations and statistical applications, including goodness-of-fit testing and inference on tail dependence under a range of dependence structures, demonstrating accurate finite-sample performance.