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

2026-01-15

AI Summary

This study addresses the challenges posed by the discreteness of empirical estimators and irregular boundary behavior in nonparametric estimation of tail copulas when marginal distributions are unknown. The authors propose a checkerboard-smoothed tail copula estimator based on local bilinear interpolation, which—combined with a multiplier bootstrap procedure—achieves consistent approximation and feasible inference for tail dependence structures without requiring explicit estimation of the limiting covariance structure. Theoretical analysis establishes the estimator’s almost sure uniform consistency and conditional weak convergence. Simulation studies confirm the accuracy and reliability of the resulting bootstrap confidence intervals, offering a novel and effective tool for inference on tail dependence coefficients.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

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.

Problem

• tail dependence
• copula
• nonparametric inference
• unknown margins
• extremal dependence

Innovation

• checkerboard copula
• tail dependence
• multiplier bootstrap
• weak convergence
• nonparametric inference