In order to understand structural relationships among sets of variables at extreme levels, we develop an extremes analogue to partial correlation. We begin by developing an inner product space constructed from transformed-linear combinations of independent regularly varying random variables. We define partial tail correlation via the projection theorem for the inner product space. We show that the partial tail correlation can be understood as the inner product of the prediction errors from transformed-linear prediction. We connect partial tail correlation to the inverse of the inner product matrix and show that a zero in this inverse implies a partial tail correlation of zero. We then show that under a modeling assumption that the random variables belong to a sensible subset of the inner product space, the matrix of inner products corresponds to the previously-studied tail pairwise dependence matrix. We develop a hypothesis test for partial tail correlation of zero. We demonstrate the performance in two applications: high nitrogen dioxide levels in Washington DC and extreme river discharges in the upper Danube basin.

翻译：为了理解极端水平下变量集之间的结构关系，我们开发了偏相关的极值类比方法。首先，我们基于独立正则变化随机变量的变换线性组合构建了一个内积空间。通过内积空间的投影定理，我们定义了尾部偏相关。研究表明，尾部偏相关可理解为变换线性预测中预测误差的内积。我们将尾部偏相关与内积矩阵的逆矩阵联系起来，并证明该逆矩阵中的零元素对应零尾部偏相关。进一步，在随机变量属于内积空间某个合理子集的建模假设下，内积矩阵对应于先前研究的尾部成对依赖矩阵。我们提出了针对零尾部偏相关的假设检验方法。通过两个应用案例验证了方法的性能：华盛顿特区高二氧化氮水平与多瑙河上游流域极端河流量。