Asymptotic Theory of Tail Dependence and Bootstrap for Checkerboard Copulas

A comprehensive asymptotic and bootstrap theory is established for checkerboard-based estimation of the copula and its lower and upper tail copula counterparts under unknown marginal distributions.

The proposed estimator of the tail copula extends a local bilinear interpolation of the empirical copula to the tail region, providing a flexible nonparametric approach for modeling extremal dependence.

Almost sure uniform consistency is established under mild conditions on the checkerboard grid.

Weak convergence of the checkerboard copula process is derived, showing that smoothing preserves the first-order asymptotic limit of the empirical copula process, including the effect of marginal estimation.

These results are further extended to lower and upper tail copula processes, yielding asymptotic normality for tail dependence measures.

Since the limiting processes depend on unknown characteristics of the underlying true copula, a multiplier bootstrap procedure adapted to the checkerboard structure is proposed and shown to be asymptotically valid.

Simulation studies and statistical applications validate our theoretical findings under a range of dependence structures.

Although the limiting processes match with that observed for the empirical copula, the finite sample performance shows a noticeable improvement under checkerboard smoothing.