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_n1 and lambda_hat_n2. 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_n2, such as its consistency, limiting distribution, and conditions under which these properties hold—key ingredients for theoretically grounded inference without resampling.

References

We proved asymptotic normality for the first estimator; deriving the asymptotic properties for the second estimator remains future work.

— The Efficient Tail Hypothesis: An Extreme Value Perspective on Market Efficiency  (2408.06661 - Jiang et al., 2024) in Section 6, Conclusion

In this paper, we have abstracted from the time-dependence of involved variables following \citet{drees2008some} that suggests that direct nonparametric methods developed for the classical i.i.d. setting can also be useful for the tail analysis of serially dependent data. However, a next logical step would be to explicitly account for temporal dependencies and apply the testing theory to the adjusted version of CTC for time-series data from the work of \citet{bodik2024causality} on data with different tails. We leave this for future work.

— Identification and Inference for Causal Effects in Extremes under General Conditions  (2608.22957 - Leimenstoll et al., 24 Aug 2026) in Section Conclusions