3,013 research outputs found
Empirical Bayesian analysis of componentwise maxima in multivariate samples
No full textStatistical theory and methods for the analysis of maxima, computed componentwise in a multivariate sample, has been an active research area in the last decade. Under mild assumptions, extreme-value theory justifies modelling random vectors of linearly normalized sample maxima by multivariate max-stable distributions. Various proposals for Bayesian inferential procedures have been formulated in recent years, though they typically disregard the asymptotic bias inherent in the use of max-stable models, incorporating no information on norming sequences in prior specifications for scale and location parameters. The semiparametric empirical Bayesian approach in Padoan and Rizzelli (2022) suitably addresses this point via data-dependent priors. In this contribution we review its consistency properties
Multivariate extreme models based on underlying skew-t and skew-normal distributions
No full textAbstractWe derive for the first time the limiting distribution of maxima of skew-t random vectors and we show that its limiting case, as the degree of freedom goes to infinity, is the skewed version of the well-known Hüsler–Reiss model. The advantage of the new families of models is that they are particularly flexible, allowing for both symmetric and asymmetric dependence structures and permitting the modelling of multivariate extremes with dimensions greater than two
Extreme dependence models based on event magnitude
No full textBy considering pointwise maxima of independent stationary random processes with dependent Cauchy marginals, we define a new process whose univariate limit distributions
are Fréchet and the bivariate distributions interpolate between independence and complete dependence. The limiting dependence structure that emerges is suitable to describe dependent margins. However, we show that it is possible to enable different levels of dependence
according to the magnitude of extreme events, e.g. the dependence decreases as the extremes’ intensity increases. In particular, with the class of random fields defined here,
the dependence of spatial extremes can be modeled. We describe some properties of the dependence structure and we illustrate its utility in assessing the dependence. Combining
marginal likelihoods through the composite likelihood approach, we are able to estimate the extremal dependence of extreme values observed in space. We convey the model’s capabilities through an analysis of sea-levels recorded along the coast of the United Kingdom
Computational Methods for Complex Problems in Extreme Value Theory
No full textRare events are part of the real world but inevitably environmental extreme events may have a massive impact on everyday life. We are familiar, for example, with the consequences and damage caused by hurricanes and floods etc. Consequently, there is considerable attention in studying, understanding and predicting the nature of such phenomena and the problems caused by them, not least because of
the possible link between extreme climate events and global warming or climate change. Thus the study of extreme events has become ever more important, both in terms of probabilistic and statistical research.
This thesis aims to provide statistical modelling and methods for making inferences about extreme events for two types of process. First, non-stationary univariate processes; second, spatial stationary processes. In each case the statistical aspects focus on model fitting and parameter estimation with applications to the
modelling of environmental processes including, in particular, nonstationary extreme
temperature series and spatially recorded rainfall measures
Analysis of Random Fields Using CompRandFld
No full textStatistical analysis based on random fields has become a widely used approach in order to better understand real processes in many fields such as engineering, environmental sciences, etc. Data analysis based on random fields can be sometimes problematic to carry out from the inferential prospective. Examples are when dealing with: large dataset, counts or binary responses and extreme values data. This article explains how to perform, with the R package CompRandFld, challenging statistical analysis based on Gaussian, binary and max-stable random fields. The software provides tools for performing the statistical inference based on the composite likelihood in complex problems where standard likelihood methods are difficult to apply. The principal features are illustrated by means of simulation examples and an application of Irish daily wind speeds
Extreme value analysis based in part on the article “Extreme value analysis” by Saralees Nadarajah, which appeared in the encyclopaedia of environmetrics.
No full textExtreme value theory concerns the behavior of the extremes of a process or processes. The fundamentals of this probability theory have been known since about the beginning of the twentieth century, but the relevant statistical methods for modeling extreme values emerged in the literature only in the past three decades. In fact, since 1980 the literature has seen a flood of applications of statistical extreme values, covering a wide range of areas. These application areas include environmental sciences, including climate, engineering and hydrology, performance assessment as in sports or policing, astronomy, finance, chemometrics, mortality studies, and outlier detection. Further references to specific applications are noted throughout this article.
The aim of this article is to review some fundamentals of extreme value theory and the relevant statistical methods. The emphasis will be on the latter and the applications it has attracted in the literature so far. The article is in two parts. The first part considers univariate extremes and the remainder is for multivariate extremes. Each part begins with a discussion of fundamental theoretical results. This is then followed by a discussion of relevant statistical models, inference, and simulation
Multivariate extremes over a random number of observations
No full textThe classical multivariate extreme-value theory concerns the modelling of extremes in a multivariate random sample, suggesting the use of max-stable distributions.
In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max-stable distributions. We also connect the extremal dependence structure of classical max-stable distributions and that of our new family of max-stable distributions. Using an inversion method, we derive a semiparametric composite-estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large-sample theory of the composite-estimator and illustrate its finite-sample performance via a simulation study
Strong Convergence of Peaks Over a Threshold
No full textExtreme Value Theory plays an important role to provide approximation results
for the extremes of a sequence of independent random variables when their
distribution is unknown. An important one is given by the {generalised Pareto
distribution} as an approximation of the distribution
of the excesses over a threshold , where is a suitable
norming function. In this paper we study the rate of convergence of
to in variational and Hellinger distances and
translate it into that regarding the Kullback-Leibler divergence between the
respective densities
Nonparametric estimation of the dependence among multivariate rainfall maxima
No full textMultivariate analysis of extreme values has an increasing range of applications in risk analysis, especially in the fields of environmental sciences. For example, it would be of interest for hydrologists to extract relevant information hidden in complex spatial-temporal rainfall datasets. The aim of this work is to analyse the dependence structures of weekly maxima of hourly rainfall in France recorded from 1993 to 2011. Some weather stations, initially organised in clusters, are analysed in order to summarise the dependence within all groups of seven stations. However, beyond the bivariate case, the analysis of the dependence structures for moderately high dimensional problems is still challenging. Estimation methods for assessing the extremal dependence must satisfy appropriate assumptions for guaranteeing valid results. The approach used here focuses on the nonparametric estimation of the Pickands dependence function through a specific type of Bernstein polynomial representation which ensures that all required constraints are verified
Max-stable processes
No full textEnvironmental problems such as floods require statistical analysis that takes into account the complex
nature of the data, namely observations are sampled at different spatial points in a given region for a certain time. Thus the spatial dependence structure cannot be ignored. Extreme statistics for the design of structures for flood protection, for the study of the structural failures such as bridges, dams, etc., for the prediction of heat waves and others should be based on a solid theoretical framework. Max-stable processes provide a such theory and in the last decade have emerged as fertile ground for research and a common tool for the statistical modeling of spatial extremes. This entry provides a summary of max-stable processes
- …
