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Journal Articles Annals of Applied Statistics Year : 2012

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

Daniel Cooley
• Function : Correspondent author
• PersonId : 949666

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Richard A. Davis
• Function : Author
Philippe Naveau

#### 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 $\mathbf{Z}=(Z_1,...,Z_d)^T$ and $\mathbf{Z}_{-d}=(Z_1,...,Z_{d-1})^T$, 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.

#### Domains

Applications [stat.AP]

### Dates and versions

cea-00917292 , version 1 (11-12-2013)

### Identifiers

• HAL Id : cea-00917292 , version 1
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### Cite

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. Annals of Applied Statistics, 2012, 6 (4), pp.1406-1429. ⟨10.1214/12-AOAS554⟩. ⟨cea-00917292⟩

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