probabilities of concurrent extremes the statistical modelling of spatial extremes has recently made major advances much of its focus so far has been on the modelling of the magnitudes of extreme events but little attention has been paid on the timing of extremes to address this gap this paper introduces the notion of extremal concurrence suppose that one measures precipitation at several synoptic stations over multiple days we say that extremes are concurrent if the maximum precipitation over time at each station is achieved simultaneously e g on a single day under general conditions we show that the finite sample concurrence probability converges to an asymptotic quantity deemed extremal concurrence probability using palm calculus we establish general expressions for the extremal concurrence probability through the max stable process emerging in the limit of the componentwise maxima of the sample explicit forms of the extremal concurrence probabilities are obtained for various max stable models and several estimators are introduced in particular we prove that the pairwise extremal concurrence probability for max stable vectors is precisely equal to the kendall s tau the estimators are evaluated by using simulations and applied to study the concurrence patterns of temperature extremes in the united states the results demonstrate that concurrence probability can provide a powerful new perspective and tools for the analysis of the spatial structure and impact of extremes

extremal behaviour of aggregated data with an application to downscaling summary the distribution of spatially aggregated data from a stochastic process x may exhibit tail behaviour different from that of its marginal distributions for a large class of aggregating functionals ell we introduce the ell extremal coefficient which quantifies this difference as a function of the extremal spatial dependence in x we also obtain the joint extremal dependence for multiple aggregation functionals applied to the same process formulae for the ell extremal coefficients and multivariate dependence structures are derived in important special cases the results provide a theoretical link between the extremal distribution of the aggregated data and the corresponding underlying process which we exploit to develop a method for statistical downscaling we apply our framework to downscale daily temperature maxima in the south of france from a gridded dataset and use our model to generate high resolution maps of the warmest day during the 2003 heatwave

modelling dependence within joint tail regions standard approaches for modelling dependence within joint tail regions are based on extreme value methods which assume max stability a particular form of joint tail dependence we develop joint tail models based on a broader class of dependence structure which provides a natural link between max stable models and weaker forms of dependence including independence and negative association this approach overcomes many of the problems that are encountered with standard methods and is the basis for a poisson process representation that generalizes existing bivariate results we apply the new techniques to simulated and environmental data and demonstrate the marked advantage that the new approach offers for joint tail extrapolation

modelling extreme multivariate events summary the classical treatment of multivariate extreme values is through componentwise ordering though in practice most interest is in actual extreme events here the point process of observations which are extreme in at least one component is considered parametric models for the dependence between components must satisfy certain constraints two new techniques for generating such models are presented aspects of the statistical estimation of the resulting models are discussed and are illustrated with an application to oceanographic data

on some types of fuzzy connectedness in fuzzy topological spaces abstract in this paper we present some counter examples related to the product of fuzzy connected spaces in addition fuzzy super c5 connectedness is defined and several relations between fuzzy c5 connectedness are obtained finally it is shown that fuzzy c5 connectedness is preserved under some types of continuity

markov chain models for threshold exceedances in recent research on extreme value statistics there has been an extensive development of threshold methods first in the univariate case and subsequently in the multivariate case as well in this paper an alternative methodology for extreme values of univariate time series is developed by assuming that the time series is markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions a new likelihood representation for threshold methods is presented which we apply to a markovian time series an important motivation for developing this kind of theory is the possibility of calculating probability distributions for functionals of extreme events we address this issue by showing how a theory of compound poisson limits for additive functionals can be combined with simulation to obtain numerical solutions for problems of practical interest the methods are illustrated by application to temperature data

the multivariate skew normal distribution summary the paper extends earlier work on the so called skew normal distribution a family of distributions including the normal but with an extra parameter to regulate skewness the present work introduces a multivariate parametric family such that the marginal densities are scalar skew normal and studies its properties with special emphasis on the bivariate case

dependence modelling for spatial extremes current dependence models for spatial extremes are based upon max stable processes within this class there are few inferentially viable models available and we propose one further model more problematic are the restrictive assumptions that must be made when using max stable processes to model dependence for spatial extremes it must be assumed that the dependence structure of the observed extremes is compatible with a limiting model that holds for all events more extreme than those that have already occurred this problem has long been acknowledged in the context of finite dimensional multivariate extremes in particular when data display dependence at observable levels but are independent in the limit we propose a flexible class of models that is suitable for such data in a spatial context in addition we consider the situation where the extremal dependence structure may vary with distance we apply our models to spatially referenced significant wave height data from the north sea finding evidence that their extremal structure is not compatible with a limiting dependence model

a mixture model for multivariate extremes this thesis is a contribution to multivariate extreme value statistics the tail of a multivariate distribution function is characterized by its spectral distribution for which we propose a new semi parametric model based on mixtures of dirichlet distributions to estimate the components of this model reversible jump monte carlo markov chain and em algorithms are developed their performances are illustrated on real and simulated data obtained using new representations of the extremal logistic and dirichlet models in parallel with the estimation of the spectral distribution extreme value statistic machinery requires the selection of a threshold in order to classify data as extreme or not this selection is achieved by a new method based on heuristic arguments it allows a selection independent of the dimension of the data its performance is illustrated on real and simulated data primal scientific interests behind a multivariate extreme value analysis reside in the estimation of quantiles of rare events and in the exploration of the dependence structure for which the estimation of the spectral measure is a means rather than an end these two issues are addressed for the first a monte carlo method is developed based on simulation of extremes it is compared with classical and new methods of the literature for the second one an original conditional dependence analysis is proposed which enlightens various aspects of the dependence structure of the data examples using real data sets are given in the last part the semi parametric model and the presented methods are extended to spatial extremes it is made possible by considering the spectral distribution as the distribution of a random probability an original viewpoint adopted throughout this thesis classical multivariate extremes are extended to extremes of random measures the application is illustrated on rainfall data in china

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

conditional simulation of max stable processes summary since many environmental processes such as heat waves or precipitation are spatial in extent it is likely that a single extreme event affects several loca tions and the areal modelling of extremes is therefore essential if the spatial dependence of e xtremes has to be appropriately taken into account this paper proposes a framework for conditional simulations of max stable processes and give closed forms for brown resnick and schlather processes we test the method on simulated data and give an application to extreme rainfall around zurich and extreme temperature in switzerland results show that the proposed framework provides accurate conditional simulations and can handle real sized problems

exact simulation of max stable processes max stable processes play an important role as models for spatial extreme events their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult algorithms based on finite approximations are often inexact and computationally inefficient we present a new algorithm for exact simulation of a max stable process at a finite number of locations it relies on the idea of simulating only the extremal functions that is those functions in the construction of a max stable process that effectively contribute to the pointwise maximum we further generalize the algorithm by dieker mikosch 2015 for brown resnick processes and use it for exact simulation via the spectral measure we study the complexity of both algorithms prove that our new approach via extremal functions is always more efficient and provide closed form expressions for their implementation that cover most popular models for max stable processes and multivariate extreme value distributions for simulation on dense grids an adaptive design of the extremal function algorithm is proposed

a new class of models for bivariate joint tails summary a fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions the primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non negative measure that must satisfy some mild constraints we then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence asymptotic independence and asymmetry within a straightforward parsimonious parameterization we provide a fast simulation algorithm for this example and detail likelihood based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection we illustrate this model by application to both simulated and real data in contrast with the classical multivariate extreme value approach which concentrates on the limiting distribution of normalized componentwise maxima our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling

modelling across extremal dependence classes different dependence scenarios can arise in multivariate extremes entailing careful selection of an appropriate class of models in bivariate extremes the variables are either asymptotically dependent or are asymptotically independent most available statistical models suit one or other of these cases but not both resulting in a stage in the inference that is unaccounted for but can substantially impact subsequent extrapolation existing modelling solutions to this problem are either applicable only on sub domains or appeal to multiple limit theories we introduce a unified representation for bivariate extremes that encompasses a wide variety of dependence scenarios and applies when at least one variable is large our representation motivates a parametric model that encompasses both dependence classes we implement a simple version of this model and show that it performs well in a range of settings

inferences on the association parameter in copula models for bivariate survival data we investigate two stage parametric and two stage semi parametric estimation procedures for the association parameter in copula models for bivariate survival data where censoring in either or both components is allowed we derive asymptotic properties of the estimators and compare their performance by simulations both parametric and semi parametric estimators of the association parameter are efficient at independence and the parameter estimates in the margins have high efficiency and are robust to misspecification of dependency structures in addition we propose a consistent variance estimator for the semi parametric estimator of the association parameter we apply the proposed methods to an aids data set for illustration

connectedness and dimension of gf spaces abstract connectedness of gf spaces defined by means of open closed sets is studied this leads to the notion of a i monotone function using connectedness the notion of a f component is introduced the relationship between connectedness and zero dimensionality is considered

the continuous and differentiable domains of attraction of the extreme value distributions on caracterise les domaines d attraction des distributions de valeurs extremes univariables en utilisant des fonctions de hasard cumulative inverses

bayesian modeling of air pollution extremes using nested multivariate max stable processes capturing the potentially strong dependence among the peak concentrations of multiple air pollutants across a spatial region is crucial for assessing the related public health risks in order to investigate the multivariate spatial dependence properties of air pollution extremes we introduce a new class of multivariate max stable processes our proposed model admits a hierarchical tree based formulation in which the data are conditionally independent given some latent nested positive stable random factors the hierarchical structure facilitates bayesian inference and offers a convenient and interpretable characterization we fit this nested multivariate max stable model to the maxima of air pollution concentrations and temperatures recorded at a number of sites in the los angeles area showing that the proposed model succeeds in capturing their complex tail dependence structure

a construction principle for multivariate extreme value distributions we present a construction principle for the spectral density of a multivariate extreme value distribution it generalizes the pairwise beta model introduced in the literature recently and may be used to obtain new parametric models from lower dimensional spectral densities we illustrate the flexibility of this new class of models and apply it to a wind speed dataset

bivariate extreme value theory models and estimation summary bivariate extreme value distributions arise as the limiting distributions of renormalized componentwise maxima no natural parametric family exists for the dependence between the marginal distributions but there are considerable restrictions on the dependence structure we consider modelling the dependence function with parametric models for which two new models are presented tests for independence and discriminating between models are also given the estimation procedure and the flexibility of the new models are illustrated with an application to sea level data extreme value theory has recently been an area of much theoretical and practical work univariate theory is a well documented area whereas bivariate multivariate extreme value theory has until recently received surprisingly little attention in the multivariate case no natural parametric family exists for the dependence structure so this must be modelled in some way in the analysis of environmental extreme value data there is a need for models of dependence between extremes from different sources for example at various sea ports or at various points of a river in this paper we consider bivariate extreme value distributions we assume without loss of generality that we have exponential marginal distributions with unit means the class of bivariate exponential distributions in which we are interested satisfy a strong stability relation exponential variables x y satisfy the stability relation if and only if w min ax by is also exponentially distributed for all a b 0 pickands 1981 therefore the models we will consider have particular application in reliability and survival analysis one approach to modelling the dependence structure is via parametric models this requires a flexible family of models which satisfy certain constraints models are of two kinds either differentiable or nondifferentiable all nondifferentiable models give distri butions which are singular with nonzero probability concentrated on a certain subspace the differentiable models have densities but the existing models are symmetric which leads to the variables being exchangeable here we present two new asymmetric differenti able models which have increased flexibility properties of the differentiable models are examined estimation of the parametric models has previously been by ad hoc methods because there is a nonregular estimation problem when the margins are independent for the

connectedness and local connectedness in fuzzy topological spaces and heyting algebra valued sets abstract in this paper we introduce four notions of local connectedness in fuzzy topological spaces the question of these notions to be good extensions is discussed and their preservation under fuzzy quotients fuzzy sums and fuzzy products is investigated moreover we introduce the notion of quasi local connectedness which is a peculiarity of the fuzzy setting

statistics for near independence in multivariate extreme values we propose a multivariate extreme value threshold model for joint tail estimation which overcomes the problems encountered with existing techniques when the variables are near independence we examine inference under the model and develop tests for independence of extremes of the marginal variables both when the thresholds are fixed and when they increase with the sample size motivated by results obtained from this model we give a new and widely applicable characterisation of dependence in the joint tail which includes existing models as special cases a new parameter which governs the form of dependence is of fundamental importance to this characterisation by estimating this parameter we develop a diagnostic test which assesses the applicability of bivariate extreme value joint tail models the methods are demonstrated through simulation and by analysing two previously published data sets

the connectedness in lowen spaces abstract a new f connectedness notion is introduced and its relation with the other connectedness notions are demonstrated furthermore the properties of f sets and f connectedness in lowen spaces are also investigated

space time modelling of extreme events max stable processes are the natural analogues of the generalized extreme value distribution when modelling extreme events in space and time under suitable conditions these processes are asymptotically justified models for maxima of independent replications of random fields and they are also suitable for the modelling of extreme measurements over high thresholds this paper shows how a pairwise censored likelihood can be used for consistent estimation of the extremes of space time data under mild mixing conditions and illustrates this by fitting an extension of a model of schlather 2002 to hourly rainfall data a block bootstrap procedure is used for uncertainty assessment estimator efficiency is considered and the choice of pairs to be included in the pairwise likelihood is discussed the proposed model fits the data better than some natural competitors

local likelihood estimation of complex tail dependence structures applied to u s precipitation extremes to disentangle the complex nonstationary dependence structure of precipitation extremes over the entire contiguous united states u s we propose a flexible local approach based on factor copula

spatial extremes models for the stationary case the aim of this paper is to provide models for spatial extremes in the case of stationarity the spatial dependence at extreme levels of a stationary process is modeled using an extension of the theory of max stable processes of de haan and pickands probab theory related fields 72 1986 477 492 we propose three one dimensional and three two dimensional models these models depend on just one parameter or a few parameters that measure the strength of tail dependence as a function of the distance between locations we also propose two estimators for this parameter and prove consistency under domain of attraction conditions and asymptotic normality under appropriate extra conditions

fuzzy connected sets in fuzzy topological spaces abstract the interrelationship between different notions of connectedness of fuzzy sets introduced by ming and ming zheng chong you saha and ajmal and kohli is investigated a new concept of connectedness called c 5 connectedness of a fuzzy set is introduced and the validity of the standard results for this new concept of connectedness is examined

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

an m estimator of spatial tail dependence tail dependence models for distributions attracted to a max stable law are tted using observations above a high threshold to cope with spatial high dimensional data a rank based m estimator is proposed relying on bivariate margins only a data driven weight matrix is used to minimize the asymptotic variance empirical process arguments show that the estimator is consistent and asymptotically normal its nite sample performance is assessed in simulation experiments involving popular max stable processes perturbed with additive noise an analysis of wind speed data from the netherlands illustrates the method

models and inference for uncertainty in extremal dependence conventionally modelling of multivariate extremes has been based on the class of multivariate extreme value distributions more recently other classes have been developed allowing for the possibility that whilst dependence is observed at finite levels the limit distribution is independent a number of articles have shown this development to be important for accurate estimation of the extremal properties both of theoretical processes and observed datasets it has also been shown that so far as dependence is concerned the choice between modelling with either asymptotically dependent or asymptotically independent distributions can be far more influential than model choice within either of these two classes in this paper we explore the issue of modelling across both classes examining in particular the effect of uncertainty caused by lack of knowledge about the status of asymptotic dependence this is achieved by new multivariate models whose parameter spaces are such that asymptotic dependence occurs on a boundary standard techniques in bayesian inference implemented through markov chain monte carlo enable inferences to be drawn that assign posterior probability mass to the boundary region the techniques are illustrated on a set of oceanographic data for which previous analyses have shown that it is difficult to resolve the question of asymptotic dependence status which is however important in model extrapolation

modeling spatial processes with unknown extremal dependence class abstractmany environmental processes exhibit weakening spatial dependence as events become more extreme well known limiting models such as max stable or generalized pareto processes cannot capture this which can lead to a preference for models that exhibit a property known as asymptotic independence however weakening dependence does not automatically imply asymptotic independence and whether the process is truly asymptotically in dependent is usually far from clear the distinction is key as it can have a large impact upon extrapolation that is the estimated probabilities of events more extreme than those observed in this work we present a single spatial model that is able to capture both dependence classes in a parsimonious manner and with a smooth transition between the two cases the model covers a wide range of possibilities from asymptotic independence through to complete dependence and permits weakening dependence of extremes even under asymptotic dependence censored likelihood based

multivariate archimedean copulas d monotone functions and l1 norm symmetric distributions it is shown that a necessary and sufficient condition for an archimedean copula generator to generate a d dimensional copula is that the generator is a d monotone function the class of d dimensional archimedean copulas is shown to coincide with the class of survival copulas of d dimensional l 1 norm symmetric distributions that place no point mass at the origin the d monotone archimedean copula generators may be characterized using a little known integral transform of williamson duke math j 23 1956 189 207 in an analogous manner to the well known bernstein widder characterization of completely monotone generators in terms of the laplace transform these insights allow the construction of new archimedean copula families and provide a general solution to the problem of sampling multivariate archimedean copulas they also yield useful expressions for the d dimensional kendall function and kendall s rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for archimedean copulas the existence of a sharp lower bound for archimedean copulas with respect to the positive lower orthant dependence ordering is shown

mapping extreme snowfalls in the french alps using max stable processes 1 the evaluation of extreme snowfalls is an important challenge for hazard management in mountainous regions in this paper the extreme snowfall data acquired from 40 meteorological stations in the french alps since 1966 are analyzed using spatial extreme statistics they are then modeled within the formal framework of max stable processes msps which are the generalization of the univariate extreme value theory to the spatial multivariate case the three main msps now available are compared using composite likelihood maximization and the most flexible brown resnick one is retained on the basis of the takeuchi information criterion taking into account anisotropy by space transformation furthermore different models with smooth trends linear and splines for the spatial evolution of the generalized extreme value gev parameters are tested to allow snowfall maps for different return periods to be produced after altitudinal correction that separates spatial and orographic effects the different spatial models are fitted to the data within the max stable framework allowing inference of the gev margins and the extremal dependence simultaneously finally a nested model selection procedure is employed to select the best linear and spline models results show that the best linear model produces reasonable quantile maps assessed by cross validation using other stations but it is outperformed by the best spline model which better captures the complex evolution of gev parameters with space for a given return period and at fixed elevation of 2000 m extreme 3 day snowfalls are higher in the ne and se of the french alps maxima of the location parameter of the gev margins are located in the north and south while maxima of the scale parameter are located in the se which corresponds to the mediterranean influence that tends to bring more variability besides the dependence of extreme snowfalls is shown to be stronger on the local orientation of the alps an important result for meteorological variables confirming previous studies computations are performed for different accumulation durations which enable obtaining magnitude frequency curves and show that the intensity of the extremal directional dependence effect is all the more important when the duration is short finally we show how the fitted model can be used to evaluate joint exceedence probabilities and conditional return level maps which can be useful for operational risk management

modelling multivariate extreme value distributions summary multivariate extreme value distributions arise as the limiting joint distribution of normalized componentwise maxima minima no parametric family exists for the depen dence between the margins this paper extends to more than two variables the models and results for the bivariate case obtained by tawn 1988 two new families of physically motivated parametric models for the dependence structure are presented and are illus trated with an application to trivariate extreme sea level data in the analysis of environmental extreme value data there is often need to study joint inter site extreme behaviour examples are joint flooding at various sea ports or at various rain gauges in this paper the class of multivariate extreme value distributions is considered the marginal distributions are all extreme value distributions without loss of generality we follow pickands s 1981 approach in transforming the margins to unit mean exponential distributions the resulting class of p variable multivariate exponential distributions satisfies a strong stability relation de haan resnick 1977 and pickands 1981 show that the resulting dependence structure is uniquely specified up to an arbitrary finite positive measure on p 1 variables therefore the bivariate dependence structure is univariate only when p 3 as considered here does the problem become really multivari ate in addition unlike in the multivariate normal distribution the multivariate depen dence structure is generally only restricted and not uniquely specified by the dependence structure of all possible pairs much has been written about the representations of multivariate extremes see resnick 1987 for a good review on the other hand models and estimation for such distributions have received little attention these aspects are considered here as no finite parametric family exists for the general dependence structure it must be modelled in some way thus true multivariate models are required using the most flexible bivariate model of tawn 1988 as a basis multivariate extensions are generated a model derived in the discrete choice literature will also be introduced and extended the resulting families of models are physically motivated flexible and possess hierarchical properties

likelihood based inference for max stable processes the last decade has seen max stable processes emerge as a common tool for the statistical modeling of spatial extremes however their application is complicated due to the unavailability of the multivariate density function and so likelihood based methods remain far from providing a complete and flexible framework for inference in this article we develop inferentially practical likelihood based methods for fitting max stable processes derived from a composite likelihood approach the procedure is sufficiently reliable and versatile to permit the simultaneous modeling of marginal and dependence parameters in the spatial context at a moderate computational cost the utility of this methodology is examined via simulation and illustrated by the analysis of united states precipitation extremes

a fuzzy topology associated with a fuzzy finite state machine abstract given a fuzzy finite state machine m q a fuzzy closure operator is defined which induces a lowen type fuzzy topology in q the concepts of separation of fuzzy subsystems and connectedness of fuzzy finite state machines are developed some results on connectedness and weak connectedness of fuzzy subsystems are obtained characterizations of these concepts in terms of the fuzzy topology are also obtained

on some fuzzy connectedness concepts abstract in this paper we examine whether or not the concepts of fuzzy strong connectedness fuzzy super connectedness and fuzzy c connectedness can be viewed as e connectedness in the category fts of fuzzy topological spaces in the sense of preuss

connectedness in fuzzy topological spaces abstract we introduce four notions of connectedness for an arbitrary fuzzy set and discuss all the implications that exist between them using these notions various equivalence relations are defined on the set of fuzzy points components and quasi components are formed as the union of certain equivalence classes a fuzzy space is shown to be a dispoint union of its components in all the four cases and non overlapping union of its quasi components

kirwan novikov inequalities on a manifold with boundary we extend the novikov morse type inequalities for closed 1 forms in 2 directions first we consider manifolds with boundary second we allow a very degenerate structure of the critical set of the form assuming only that the form is non degenerated in the sense of kirwan in particular we obtain a generalization of a result of floer about the usual morse inequalities on a manifold with boundary we also obtain an equivariant version of our inequalities our proof is based on an application of the witten deformation technique the main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end this leads to a considerable simplification of the local analysis in particular we obtain a new analytic proof of the morse bott inequalities on a closed manifold

quotient correlation a sample based alternative to pearson s correlation the quotient correlation is defined here as an alternative to pearson s correlation that is more intuitive and flexible in cases where the tail behavior of data is important it measures nonlinear dependence where the regular correlation coefficient is generally not applicable one of its most useful features is a test statistic that has high power when testing nonlinear dependence in cases where the fisher s z transformation test may fail to reach a right conclusion unlike most asymptotic test statistics which are either normal or 2 this test statistic has a limiting gamma distribution henceforth the gamma test statistic more than the common usages of correlation the quotient correlation can easily and intuitively be adjusted to values at tails this adjustment generates two new concepts the tail quotient correlation and the tail independence test statistics which are also gamma statistics due to the fact that there is no analogue of the correlation coefficient in extreme value theory and there does not exist an efficient tail independence test statistic these two new concepts may open up a new field of study in addition an alternative to spearman s rank correlation a rank based quotient correlation is also defined the advantages of using these new concepts are illustrated with simulated data and a real data analysis of internet traffic

geostatistics of extremes we describe a prototype approach to flexible modelling for maxima observed at sites in a spatial domain based on fitting of max stable processes derived from underlying gaussian random fields the models we propose have generalized extreme value marginal distributions throughout the spatial domain consistent with statistical theory for maxima in simpler cases and can incorporate both geostatistical correlation functions and random set components parameter estimation and fitting are performed through composite likelihood inference applied to observations from pairs of sites with occurrence times of maxima taken into account if desired and competing models are compared using appropriate information criteria diagnostics for lack of model fit are based on maxima from groups of sites the approach is illustrated using annual maximum temperatures in switzerland with risk analysis proposed using simulations from the fitted max stable model drawbacks and possible developments of the approach are discussed

t product fuzzy topology abstract using the notion of a t norm the t product fuzzy topological space of two fuzzy topological spaces is defined the validity of some standard results on product fuzzy topology defined by wong 10 is examined in the case of t product fuzzy topology

factor copula models for replicated spatial data abstractwe propose a new copula model that can be used with replicated spatial data unlike the multivariate normal copula the proposed copula is based on the assumption that a common factor exists and affects the joint dependence of all measurements of the process moreover the proposed copula can model tail dependence and tail asymmetry the model is parameterized in terms of a covariance function that may be chosen from the many models proposed in the literature such as the matern model for some choice of common factors the joint copula density is given in closed form and therefore likelihood estimation is very fast in the general case one dimensional numerical integration is needed to calculate the likelihood but estimation is still reasonably fast even with large datasets we use simulation studies to show the wide range of dependence structures that can be generated by the proposed model with different choices of common factors we apply the proposed model to spatial temperature data and com

overlapping families and covering dimension in fuzzy topological spaces abstract a non classical approach is used to introduce notions of overlapping families of fuzzy sets and the order of a family of fuzzy sets leading to a coherent theory of covering dimension in fuzzy topological spaces class of semicrisp fuzzy topological spaces is introduced which provides a suitable framework for the study of dimension function in the fuzzy setting

connectedness in l fuzzy topological spaces based on the consideration to the layer structures of fuzzy lattices and the level topologies of l fuzzy topological spaces a connectedness is defined for an arbitrary l fuzzy set in this paper this definition reflects the degree of connectivity of an l fuzzy set it is shown that the connectedness of l fuzzy topological spaces is an l good extension multiplicative and preserved under continuous l valued zadeh functions and that the inverse limit of continuums is a continuum general l fuzzy intervals and h intervals are defined and connectedness of them are proved it is also shown that the connectedness of an l fuzzy topological space is equivalent to the connectedness of its induced i l fuzzy topological space

stationary max stable fields associated to negative definite functions let wi i be independent copies of a zero mean gaussian process w t t d with stationary increments and variance 2 t independently of wi let i 1 ui be a poisson point process on the real line with intensity e y dy we show that the law of the random family of functions vi i where vi t ui wi t 2 t 2 is translation invariant in particular the process t i 1 vi t is a stationary max stable process with standard gumbel margins the process arises as a limit of a suitably normalized and rescaled pointwise maximum of n i i d stationary gaussian processes as n if and only if w is a nonisotropic fractional brownian motion on d under suitable conditions on w the process has a mixed moving maxima representation

estimation of husler reiss distributions and brown resnick processes summary estimation of extreme value parameters from observations in the max domain of attraction of a multivariate max stable distribution commonly uses aggregated data such as block maxima multivariate peaks over threshold methods in contrast exploit additional information from the non aggregated large observations we introduce an approach based on peaks over thresholds that provides several new estimators for processes in the max domain of attraction of the frequently used husler reiss model and its spatial extension brown resnick processes the method relies on increments t0 conditional on t0 exceeding a high threshold where t0 is a fixed location when the marginals are standardized to the gumbel distribution these increments asymptotically form a gaussian process resulting in computationally simple estimates of the husler reiss parameter matrix and particularly enables parametric inference for brown resnick processes based on high dimensional multivariate densities this is a major advantage over composite likelihood methods that are commonly used in spatial extreme value statistics since they rely only on bivariate densities a simulation study compares the performance of the new estimators with other commonly used methods as an application we fit a non isotropic brown resnick process to the extremes of 12 year data of daily wind speed measurements

a conditional approach for multivariate extreme values with discussion summary multivariate extreme value theory and methods concern the characterization estimation and extrapolation of the joint tail of the distribution of a d dimensional random variable existing approaches are based on limiting arguments in which all components of the variable become large at the same rate this limit approach is inappropriate when the extreme values of all the variables are unlikely to occur together or when interest is in regions of the support of the joint distribution where only a subset of components is extreme in practice this restricts existing methods to applications where d is typically 2 or 3 under an assumption about the asymptotic form of the joint distribution of a d dimensional random variable conditional on its having an extreme component we develop an entirely new semiparametric approach which overcomes these existing restrictions and can be applied to problems of any dimension we demonstrate the performance of our approach and its advantages over existing methods by using theoretical examples and simulation studies the approach is used to analyse air pollution data and reveals complex extremal dependence behaviour that is consistent with scientific understanding of the process we find that the dependence structure exhibits marked seasonality with ex tremal dependence between some pollutants being significantly greater than the dependence at non extreme levels

bivariate threshold methods for extremes summary an extension of the threshold method for extreme values is developed to consider the joint distribution of extremes of two variables the methodology is based on the point process representation of bivariate extremes both parametric and nonparametric models are considered the simplest case to handle is that in which both marginal distributions are known for the more realistic case in which the marginal distributions are unknown a mixed parametric nonparametric method is proposed the techniques are illustrated with data on sulphate and nitrate levels taken from a major study of acid rain

a dependence measure for multivariate and spatial extreme values properties and inference we present properties of a dependence measure that arises in the study of extreme values in multivariate and spatial problems for multivariate problems the dependence measure characterises dependence at the bivariate level for all pairs and all higher orders up to and including the dimension of the variable necessary and sufficient conditions are given for subsets of dependence measures to be self consistent that is to guarantee the existence of a distribution with such a subset of values for the dependence measure for pairwise dependence these conditions are given in terms of positive semidefinite matrices and non differentiable positive definite functions we construct new nonparametric estimators for the dependence measure which unlike all naive nonparametric estimators impose these self consistency properties as the new estimators provide an improvement on the naive methods both in terms of the inferential and interpretability properties their use in exploratory extreme value analyses should aid the identification of appropriate dependence models the methods are illustrated through an analysis of simulated multivariate data which shows that a lack of self consistency is frequently a problem with the existing estimators and by a spatial analysis of daily rainfall extremes in south west england which finds a smooth decay in extremal dependence with distance

efficient inference and simulation for elliptical pareto processes recent advances in extreme value theory have established ell pareto processes as the natural limits for extreme events defined in terms of exceedances of a risk functional here we provide methods for the practical modelling of data based on a tractable yet flexible dependence model we introduce the class of elliptical ell pareto processes which arise as the limit of threshold exceedances of certain elliptical processes characterized by a correlation function and a shape parameter an efficient inference method based on maximizing a full likelihood with partial censoring is developed novel procedures for exact conditional and unconditional simulation are proposed these ideas are illustrated using precipitation extremes in switzerland

zero dimensional and strongly zero dimensional fuzzy topological spaces abstract concepts of zero dimensionality strong zero dimensionality and total disconnectedness are extended to fuzzy topological spaces their basic properties and interrelationships among them are studied a sum theorem for strongly zero dimensional fuzzy topological spaces is also obtained

diagnostics for dependence within time series extremes summary the analysis of extreme values within a stationary time series entails various assumptions concerning its long and short range dependence we present a range of new diagnostic tools for assessing whether these assumptions are appropriate and for identifying structure within extreme events these tools are based on tail characteristics of joint survivor functions but can be implemented by using existing estimation methods for extremes of univariate independent and identically distributed variables our diagnostic aids are illustrated through theoretical examples simulation studies and by application to rainfall and exchange rate data on the basis of these diagnostics we can explain characteristics that are found in the observed extreme events of these series and also gain insight into the properties of events that are more extreme than those observed

efficient inference for spatial extreme value processes associated to log gaussian random functions 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

a moment estimator for the index of an extreme value distribution on generalise l estimateur bien connu de hill de l indice d une fonction de reparatition avec queue de variation reguliere a une estimation de l indice d une loi de valeurs extremes on demontre la convergence et la normalite asymptotique on utilise l estimateur pour certaines estimations comme celle d une quantile elevee et d un point d extremite

the stationary bootstrap abstract this article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations previously a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m dimensional joint distribution of m consecutive observations where m is fixed this procedure has been generalized by constructing a blocks of blocks resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole i e infinite dimensional joint distribution of the stationary sequence of observations these methods share the construction of resampling blocks of observations to form a pseudo time series so that the statistic of interest may be recalculated based on the resampled data set but in the context of applying this method to stationary data it is natural

on the likelihood function of gaussian max stable processes summary wederiveaclosedformexpressionforthelikelihoodfunctionofagaussianmax stableprocessindexed by r d at p d 1 sites d 1 we demonstrate the gain in efficiency in the maximum composite likelihood estimators of the covariance matrix from p 2t op 3 sites in r 2 by means of a monte carlo