随机删失下多元极值的统计
Statistics of multivariate extremes under random censoring
- University of Copenhagen（哥本哈根大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出在随机右删失下估计多元极值尾部依赖性的方向Kaplan-Meier估计器，无需多元生存函数，证明其一致性和弱收敛，并应用于洪水保险数据。
AI中文摘要：
我们研究了d维随机向量的尾部依赖性，其坐标受到随机右删失的影响。沿每个固定方向，删失问题精确地简化为一元问题，观测数据决定了半径以及它是由事件向量还是删失向量产生的。因此，普通的Kaplan-Meier乘积限估计了该方向的联合尾部概率，适用于任意维度，且无需多元生存函数、无需平滑、除阈值外无需调整参数。该方向估计器的理论在任意边际标准化下制定，条件仅施加于标准化分布，特别适用于任意最大吸引域。我们证明了在有效联合极值数量的平方根速率下的一致性和函数弱收敛性，允许标准化被估计。乘法标准化恢复了重尾理论，而基于边际（非方向）Kaplan-Meier估计器构建的标准化不需要边际尾部模型，直接针对归一化尾部copula。对于前者，标准化误差可忽略；对于后者，在联合删失的温和条件下也可忽略。模拟研究验证了估计器的有限样本性能。对包含飓风Ian产生的索赔的国家洪水保险计划数据集的应用，同时估计了建筑和内容损失在赔偿金（受上限限制）下的联合上尾。
英文摘要：
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.
