The Annals of Applied Statistics

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

Daniel Cooley, Richard A. Davis, and Philippe Naveau

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Phenomena such as air pollution levels are of greatest interest when observations are large, but standard prediction methods are not specifically designed for large observations. We propose a method, rooted in extreme value theory, which approximates the conditional distribution of an unobserved component of a random vector given large observed values. Specifically, for $\mathbf{Z}=(Z_{1},\ldots,Z_{d})^{T}$ and $\mathbf{Z}_{-d}=(Z_{1},\ldots,Z_{d-1})^{T}$, the method approximates the conditional distribution of $[Z_{d}|\mathbf{Z}_{-d}=\mathbf{z}_{-d}]$ when $\|\mathbf{z}_{-d}\|>r_{*}$. The approach is based on the assumption that $\mathbf{Z}$ is a multivariate regularly varying random vector of dimension $d$. The conditional distribution approximation relies on knowledge of the angular measure of $\mathbf{Z}$, which provides explicit structure for dependence in the distribution’s tail. As the method produces a predictive distribution rather than just a point predictor, one can answer any question posed about the quantity being predicted, and, in particular, one can assess how well the extreme behavior is represented.

Using a fitted model for the angular measure, we apply our method to nitrogen dioxide measurements in metropolitan Washington DC. We obtain a predictive distribution for the air pollutant at a location given the air pollutant’s measurements at four nearby locations and given that the norm of the vector of the observed measurements is large.

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Ann. Appl. Stat., Volume 6, Number 4 (2012), 1406-1429.

First available in Project Euclid: 27 December 2012

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Multivariate regular variation threshold exceedances angular or spectral measure air pollution nitrogen dioxide monitoring


Cooley, Daniel; Davis, Richard A.; Naveau, Philippe. Approximating the conditional density given large observed values via a multivariate extremes framework, with application to environmental data. Ann. Appl. Stat. 6 (2012), no. 4, 1406--1429. doi:10.1214/12-AOAS554.

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