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Approximating the conditional density given large observed values via a multivariate extremes framework, with application to environmental data. (English) Zbl 1257.62118

Summary: 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 \(\mathbf {Z}=(Z_{1},\dots,Z_{d})^{T}\) and \(\mathbf {Z}_{-d}=(Z_{1},\dots,Z_{d-1})^{T}\), the method approximates the conditional distribution of \([Z_{d}|\mathbf {Z}_{-d}=\mathbf {z}_{-d}]\) when \(\parallel \mathbf {z}_{-d}\parallel >r_{\ast}\). 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.

MSC:

62P12 Applications of statistics to environmental and related topics
62H10 Multivariate distribution of statistics
62G32 Statistics of extreme values; tail inference

Software:

ismev; evd
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References:

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