Nonparametric estimation and inference for lower and upper tail copulas under unknown marginal distributions are considered. To mitigate the inherent discreteness and boundary irregularities of the empirical tail copula, a checkerboard smoothed tail copula estimator based on local bilinear interpolation is introduced. Almost sure uniform consistency and weak convergence of the centered and scaled empirical checkerboard tail copula process are established in the space of bounded functions. The resulting Gaussian limit differs from its known-marginal counterpart and incorporates additional correction terms that account for first-order stochastic errors arising from marginal estimation. Since the limiting covariance structure depends on the unknown tail copula and its partial derivatives, direct asymptotic inference is generally infeasible. To address this challenge, a direct multiplier bootstrap procedure tailored to the checkerboard tail copula is developed. By combining multiplier reweighting with checkerboard smoothing, the bootstrap preserves the extremal dependence structure of the data and consistently captures both joint tail variability and the effects of marginal estimation. Conditional weak convergence of the bootstrap process to the same Gaussian limit as the original estimator is established, yielding asymptotically valid inference for smooth functionals of the tail copula, including the lower and upper tail dependence coefficient. The proposed approach provides a fully feasible framework for confidence regions and hypothesis testing in tail dependence analysis without requiring explicit estimation of the limiting covariance structure. A simulation study illustrates the finite-sample performance of the proposed estimator and demonstrates the accuracy and reliability of the bootstrap confidence intervals under various dependence structures and tuning parameter choices.

翻译：本文研究了在边缘分布未知条件下，非参数估计与推断方法在上下尾部连接函数中的应用。为缓解经验尾部连接函数固有的离散性与边界不规则性，提出了一种基于局部双线性插值的棋盘式平滑尾部连接函数估计量。在有界函数空间中，建立了中心化与尺度化经验棋盘尾部连接函数过程的几乎必然一致收敛性与弱收敛性。所得高斯极限与已知边缘分布情形下的经典结果不同，其包含了额外的校正项以处理边缘估计所产生的一阶随机误差。由于极限协方差结构依赖于未知的尾部连接函数及其偏导数，直接进行渐近推断通常不可行。为解决此问题，本文专门针对棋盘尾部连接函数设计了一种直接乘数自助法。该方法通过将乘数重加权与棋盘平滑相结合，保留了数据的极值相依结构，并一致地捕捉了联合尾部变异性和边缘估计的影响。建立了自助法过程条件弱收敛于与原估计量相同的高斯极限，从而为尾部连接函数的平滑泛函（包括上下尾部相依系数）提供了渐近有效的推断框架。所提方法为尾部相依分析中的置信域构建与假设检验提供了完全可行的框架，无需显式估计极限协方差结构。模拟研究展示了所提估计量在有限样本下的表现，并验证了在不同相依结构与调节参数选择下自助置信区间的准确性与可靠性。