Asymptotic properties of the second estimator for directional tail dependence

Derive the asymptotic properties of the estimator lambda_hat_n^2 for the directional tail dependence measure lambda(X,Y) for bivariate balanced regularly varying random vectors, including establishing formal large-sample behavior under the modeling framework introduced in the paper.

Background

The paper introduces the directional tail dependence (DTD) measure lambda(X,Y) for bivariate balanced regularly varying vectors and proposes two estimators: lambda_hat_n^1 and lambda_hat_n^2. The first estimator’s asymptotic normality is established via a central limit theorem, enabling a t-test under suitable thresholding. However, the asymptotic properties of the second estimator are not provided; instead, the authors rely on a permutation test for inference.

This leaves unresolved the formal large-sample theory for lambda_hat_n^2, such as its consistency, limiting distribution, and conditions under which these properties hold—key ingredients for theoretically grounded inference without resampling.