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Phenomena such as air pollution levels are of greatest interest whenobservations are large, but standard prediction methods are not specificallydesigned for large observations. We propose a method, rooted in extreme valuetheory, which approximates the conditional distribution of an unobservedcomponent of a random vector given large observed values. Specifically, for$\mathbf{Z}=(Z_1,...,Z_d)^T$ and $\mathbf{Z}_{-d}=(Z_1,...,Z_{d-1})^T$, themethod approximates the conditional distribution of$[Z_d|\mathbf{Z}_{-d}=\mathbf{z}_{-d}]$ when $|\mathbf{z}_{-d}|>r_*$. Theapproach is based on the assumption that $\mathbf{Z}$ is a multivariateregularly varying random vector of dimension $d$. The conditional distributionapproximation relies on knowledge of the angular measure of $\mathbf{Z}$, whichprovides explicit structure for dependence in the distribution's tail. As themethod produces a predictive distribution rather than just a point predictor,one can answer any question posed about the quantity being predicted, and, inparticular, one can assess how well the extreme behavior is represented. Usinga fitted model for the angular measure, we apply our method to nitrogen dioxidemeasurements in metropolitan Washington DC. We obtain a predictive distributionfor the air pollutant at a location given the air pollutant's measurements atfour nearby locations and given that the norm of the vector of the observedmeasurements is large.

Approximating the conditional density given large observed values via a multivariate extremes framework, with application to environmental data