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

The linking of two models through a summary measure implies an angular measure mismatch. We expect this feature to allow for better fitting to real data that do not have discrete angular measures but the mathematical implications are not well understood.

— A Proxy-likelihood Estimator for Multivariate Extremes Models with Intractable Likelihoods  (2609.30244 - Wixson et al., 24 Sep 2026) in Discussion, Section 7

Three questions are left open for future work.

Efficiency. No efficiency claim is made here. For the full multivariate survival function under censoring the nonparametric maximum likelihood estimator is inefficient \citep{GillVanDerLaanWellner1995,vanderLaan1996}. However, the directional product limit uses, at each $q$, the one-dimensional reduction of the data. Whether the resulting asymptotic variance attains the nonparametric bound for $F\circ$ is unknown. The boundary case is settled, since without censoring, i.e., when $\Delta_{i,n}\equiv1$, the product limit KM is the empirical survival function of the reduced sample, and estimator is the empirical joint-exceedance estimator, whose optimality is classical, see \citet{DreesHuang1998}.

Identifying cases where the standardization contributes. When the regularity of Remark~\ref{rem:regular} is not implied by Assumption~\ref{a:pluginnorm}, the process $\mathbb M$ requires identification for specific standardizations. For instance, we showed that it is equal to zero for the multiplicative standardization. However, in some instances, for the Kaplan--Meier standardization, a joint weak limit of the marginal processes and the directional one is required, as well as an asymptotic expansion of the displacement of the pair $(W_{i,n}(q),\Delta_{i,n}(q))$ under the estimated standardization; the indicator and the radius move together at a near-tie, so the two cannot be separated.

Asymptotic independence. Extending the theory to the near-independence regime of \citet{LedfordTawn1996} and to hidden regular variation \citep{Resnick2002}, where $p_n(q)$ decays at a rate slower than the one $r_n$ is built from, requires a different scaling regime, but otherwise seems promising under similar techniques, albeit under additional conditions.

— Statistics of multivariate extremes under random censoring  (2609.37904 - Bladt, 29 Sep 2026) in Section 6, Discussion and outlook, paragraph headed “Asymptotic independence”