The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In this paper, we test the goodness-of-fit of a given parametric model to the extremal dependence structure of a bivariate random sample. The proposed test statistic consists of a weighted $L_1$-Wasserstein distance between a nonparametric, rank-based estimator of the true angular measure obtained by maximizing a Euclidean likelihood on the one hand, and a parametric estimator of the angular measure on the other hand. The asymptotic distribution of the test statistic under the null hypothesis is derived and is used to obtain critical values for the proposed testing procedure via a parametric bootstrap. Consistency of the bootstrap algorithm is proved. A simulation study illustrates the finite-sample performance of the test for the logistic and H\"usler--Reiss models. We apply the method to test for the H\"usler--Reiss model in the context of river discharge data.

翻译：单位球面上的角测度刻画了随机向量各分量在极端区域的一阶相依结构，其定义基于标准化边缘分布。对该测度的统计复原是处理远离数据中心观测值的学习问题中的重要步骤。本文针对二元随机样本的极值相依结构，检验给定参数模型的拟合优度。所提出的检验统计量由两部分构成：一方面是通过最大化欧几里得似然得到的非参数秩估计量（用于估计真实角测度），另一方面是角测度的参数估计量，两者之间的加权$L_1$-Wasserstein距离即为检验统计量。推导了原假设下检验统计量的渐近分布，并通过参数自助法获得检验程序的临界值。证明了自助算法的一致性。模拟研究展示了该方法在logistic模型和Hüsler-Reiss模型中的有限样本表现。最后将该方法应用于河流流量数据，对Hüsler-Reiss模型进行检验。