Partial Tail Correlation for Extremes

We develop a method for investigating conditional extremal relationships between variables at their extreme levels. We consider an inner product space constructed from transformed-linear combinations of independent regularly varying random variables. By developing the projection theorem for the inner product space, we derive the concept of partial tail correlation via projection theorem. We show that the partial tail correlation can be understood as the inner product of the prediction errors associated with the best transformed-linear prediction. Similar to Gaussian cases, 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 develop a hypothesis test for the partial tail correlation of zero and demonstrate the performance in a simulation study as well as in two applications: high nitrogen dioxide levels in Washington DC and extreme river discharges in the upper Danube basin.