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December 2012 Approximating the conditional density given large observed values via a multivariate extremes framework, with application to environmental data
Daniel Cooley, Richard A. Davis, Philippe Naveau
Ann. Appl. Stat. 6(4): 1406-1429 (December 2012). DOI: 10.1214/12-AOAS554

Abstract

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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Daniel Cooley. Richard A. Davis. Philippe Naveau. "Approximating the conditional density given large observed values via a multivariate extremes framework, with application to environmental data." Ann. Appl. Stat. 6 (4) 1406 - 1429, December 2012. https://doi.org/10.1214/12-AOAS554

Information

Published: December 2012
First available in Project Euclid: 27 December 2012

zbMATH: 1257.62118
MathSciNet: MR3058669
Digital Object Identifier: 10.1214/12-AOAS554

Keywords: Air pollution , angular or spectral measure , multivariate regular variation , nitrogen dioxide monitoring , threshold exceedances

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.6 • No. 4 • December 2012
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