Multivariate normal distribution

Summary

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Definitions Notation and parameterization The multivariate normal distribution of a k-dimensional random vector \mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T} can be written in the following notation: : \mathbf{X}\ \sim\ \mathcal{N}(\boldsymbol\mu,, \boldsymbol\Sigma),

Official source

https://en.wikipedia.org/wiki/Multivariate_normal_distribution

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EPFL2017

Efficient inference for spatial extreme value processes associated to log-Gaussian random functions

Jonathan A. Tawn, Jennifer Lynne Wadsworth

Max-stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed stochastic processes, and thus form an important class of models for the extreme values of spatial processes. Until recently, inference for max-stable processes has been restricted to the use of pairwise composite likelihoods, due to intractability of higher-dimensional distributions. In this work we consider random fields that are in the domain of attraction of a widely used class of max-stable processes, namely those constructed via manipulation of log-Gaussian random functions. For this class, we exploit limiting d-dimensional multivariate Poisson process intensities of the underlying process for inference on all d-vectors exceeding a high marginal threshold in at least one component, employing a censoring scheme to incorporate information below the marginal threshold. We also consider the d-dimensional distributions for the equivalent max-stable process, and perform full likelihood inference by exploiting the methods of Stephenson & Tawn (2005), where information on the occurrence times of extreme events is shown to dramatically simplify the likelihood. The Stephenson-Tawn likelihood is in fact simply a special case of the censored Poisson process likelihood. We assess the improvements in inference from both methods over pairwise likelihood methodology by simulation.

Oxford University Press2014

A new representation for multivariate tail probabilities

Jonathan A. Tawn, Jennifer Lynne Wadsworth

Existing theory for multivariate extreme values focuses upon characterizations of the distributional tails when all components of a random vector, standardized to identical margins, grow at the same rate. In this paper, we consider the effect of allowing the components to grow at different rates, and characterize the link between these marginal growth rates and the multivariate tail probability decay rate. Our approach leads to a whole class of univariate regular variation conditions, in place of the single but multivariate regular variation conditions that underpin the current theories. These conditions are indexed by a homogeneous function and an angular dependence function, which, for asymptotically independent random vectors, mirror the role played by the exponent measure and Pickands' dependence function in classical multivariate extremes. We additionally offer an inferential approach to joint survivor probability estimation. The key feature of our methodology is that extreme set probabilities can be estimated by extrapolating upon rays emanating from the origin when the margins of the variables are exponential. This offers an appreciable improvement over existing techniques where extrapolation in exponential margins is upon lines parallel to the diagonal.

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