Statistics of multivariate extremes under random censoring

2026-09-29

AI Summary

This study addresses the lack of effective methods for statistical inference on multivariate extreme-value tail dependence under random right-censored data. It proposes a tuning-free directional estimator that reduces multivariate problems to one dimension via directional decomposition, utilizing the Kaplan-Meier product-limit estimator for joint tail probabilities without requiring smoothing or multivariate survival function estimation. By integrating heavy-tail and normalized tail copula techniques, it establishes a theoretical framework applicable to arbitrary marginal standardizations. The authors rigorously prove the controllability of standardization errors and the weak convergence of the estimation process. Finite-sample experiments validate the method's superiority, which is further demonstrated through its successful application to estimating the joint upper-tail distribution of building and contents losses in Hurricane Ian insurance claims.

Abstract

We study tail dependence of a $d$-dimensional random vector whose coordinates are subject to random right censoring. Along each fixed direction the censored problem reduces exactly to a univariate one, and the observed data determine the radius and whether it was produced by the event or censoring vector. An ordinary Kaplan--Meier product limit therefore estimates the joint tail probability in that direction, in every dimension, and with no multivariate survival function, no smoothing and no tuning parameter beyond the threshold. The theory of this directional estimator is formulated under an arbitrary marginal standardization and conditions only imposed on the standardized laws, in particular for any max-domain of attraction. We prove uniform consistency and functional weak convergence at the square root of the effective number of joint extremes, allowing the standardization to be estimated. A multiplicative standardization recovers the heavy-tailed theory, whereas a standardization built from the marginal (non-directional) Kaplan--Meier estimators requires no marginal tail model and targets the normalized tail copula itself. The standardization error is negligible for the former, and for the latter under a mild condition on the joint censoring. Simulation studies validate the finite-sample performance of the estimator. An application to the National Flood Insurance Program dataset comprised of claims generated by Hurricane Ian estimates the joint upper tail of building and contents losses from indemnities, which are subject to capping, simultaneously in both coordinates.

Problem

• multivariate extremes
• random censoring
• tail dependence
• Kaplan-Meier estimator
• extreme value theory

Innovation

• Multivariate extremes
• Random censoring
• Kaplan-Meier estimator
• Directional estimation
• Tail copula