## 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

ismev; evd
Full Text:

### References:

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