Asymptotic Theory of Tail Dependence Measures for Checkerboard Copula and the Validity of Multiplier Bootstrap
Mayukh Choudhury, Debraj Das, Sujit Ghosh
TL;DR
The paper develops a comprehensive asymptotic theory for checkerboard-based tail copula estimation under unknown margins, introducing a bilinear-interpolation checkerboard estimator that yields a smooth, consistent approximation to the copula and its tail functionals. It proves almost sure consistency and a Gaussian limit for the copula process, derives weak convergence for the lower and upper tail copulas with marginal-estimation corrections, and shows strong consistency for the tail copulas. To enable feasible inference without explicit covariance estimation, the authors devise a direct multiplier bootstrap that combines multiplier reweighting with checkerboard smoothing, and prove conditional convergence to the same Gaussian limit for smooth functionals like the tail dependence coefficients. A simulation study demonstrates favorable finite-sample performance, confirming accurate bootstrap confidence intervals and robust behavior across dependence structures and tuning parameter choices.
Abstract
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.