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.

翻译：本文在边际分布未知的情况下，发展了基于棋盘型插值的下尾与上尾连接函数估计的完整渐近与自助法理论。该估计量通过经验连接函数的局部双线性（棋盘型）插值构造，并扩展至尾部区域，从而获得极值依赖的非参数估计。我们首先通过将误差分解为随机经验过程项与棋盘投影引起的确定性近似偏差，建立了棋盘平滑连接函数估计量的几乎必然一致相合性。在网格大小的温和增长条件下，该估计量被证明具有强相合性。接着，我们在$\ell^\infty([0,1]^2)$空间中推导了中心化与尺度化棋盘连接函数过程的弱收敛性，表明平滑过程不影响一阶极限。由此得到的高斯过程与经验连接函数的过程一致，但额外增加了源于边际估计的项。这些结果可推广至下尾与上尾连接函数过程，从而获得函数中心极限定理以及尾部相依系数的渐近正态性。由于极限协方差依赖于未知的尾部特征与偏导数，导致直接推断不可行，我们提出了一种适用于棋盘结构的直接乘子自助法。我们证明了自助法过程的条件弱收敛于同一极限，从而确保光滑泛函的有效推断。最后，通过模拟与统计应用（包括拟合优度检验与不同相依结构下尾部依赖的推断）展示了该自助法方法，验证了其准确的有限样本表现。