a nonparametric estimation procedure for bivariate extreme value copulas summary a bivariate extreme value distribution with fixed marginals is generated by a onedimensional map called a dependence function this paper proposes a new nonparametric estimator of this function its asymptotic properties are examined and its small sample behaviour is compared to that of other rank based and likelihood based procedures the new estimator is shown to be uniformly strongly convergent and asymptotically unbiased through simulations it is also seen to perform reasonably well against the maximum likelihood estimator based on the correct model and to have smaller l1 l2 and l errors than any existing nonparametric alternative the n consistency of the proposed estimator leads to nonparametric estimation of tawn s 1988 dependence measure that may be used to test independence in small samples

on a family of linear maps from mn c to mn2 c abstract bhat 1 characterizes the family of linear maps defined on b h which preserve unitary conjugation we generalize this idea and study the maps with a similar equivariance property on finite dimensional matrix algebras we show that the maps with equivariance property are significant to study k positivity of linear maps defined on finite dimensional matrix algebras in 5 choi showed that n positivity is different from n 1 positivity for the linear maps defined on n by n matrix algebras in this paper we present a parametric family of linear maps n m n c m n 2 c and study the properties of positivity completely positivity decomposability etc we determine values of parameters and for which the family of maps n is positive for any natural number n 3 we focus on the case of n 3 that is 3 and study the properties of 2 positivity completely positivity and decomposability in particular we give values of parameters and for which the family of maps 3 is 2 positive and not completely positive

lie symmetries of a coupled nonlinear burgers heat equation system a symmetry group analysis of a coupled burgers heat equation system of partial differential equations initially introduced in a study of non classical similarity solutions of the heat equation is carried out the point lie algebra of the system g is shown to possess a maximal solvable ideal a with quotient algebra b g a approximately sl2 r where sl2 r is the lie algebra of 2 2 matrices with zero trace analysis of the prolongation structure of the system yields the backlund transformation obtained previously by painleve analysis the backlund transformation can be expressed as a map onto two coupled linear heat equations or alternatively as a map onto burgers equation the role of sl2 r in a seven dimensional lie algebra l1 obtained by truncating the open ended algebraic prolongation structure is emphasised

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

structure of irreducibly covariant quantum channels for finite groups we obtain an explicit characterization of linear maps in particular quantum channels which are covariant with respect to an irreducible representation u of a finite group g whenever u uc is simply reducible with uc being the contragradient representation using the theory of group representations we obtain the spectral decomposition of any such linear map the eigenvalues and orthogonal projections arising in this decomposition are expressed entirely in terms of representation characteristics of the group g this in turn yields necessary and sufficient conditions on the eigenvalues of any such linear map for it to be a quantum channel we also obtain a wide class of quantum channels which are irreducibly covariant by construction for two dimensional irrreducible representations of the symmetric group s 3 and the quaternion group q we also characterize quantum channels which are both irreducibly covariant and entanglement breaking

rank based inference for bivariate extreme value copulas consider a continuous random pair x y whose dependence is characterized by an extreme value copula with pickands dependence function a when the marginal distributions of x and y are known several consistent estimators of a are available most of them are variants of the estimators due to pickands bull inst internat statist 49 1981 859 878 and cap era a foug eres and genest biometrika 84 1997 567 577 in this paper rank based versions of these estimators are proposed for the more common case where the margins of x and y are unknown results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank based estimators their finite and large sample performance is then compared to that of their known margin analogues as well as with endpoint corrected versions thereof explicit formulas and consistent estimates for their asymptotic variances are also suggested

l measure of non exchangeability for bivariate extreme value and archimax copulas in the class of bivariate extreme value copulas an upper bound is calculated for the measure of non exchangeability based on the l norm distance between a copula c and its transpose ct x y c y x copulas that are maximally non exchangeable with respect to are also determined moreover similar upper bounds are given respectively for the class of all ev copulas having a fixed upper tail dependence coefficient and for the larger class of archimax copulas

on the geometry of positive maps in matrix algebras ii abstract geometric relations among the identity map the transpose map and the map x 1 n x tr x 1 n in the matrix algebra mn are illustrated providing parametric examples of maps which are not k 1 positive but which satisfy the matrix schwartz inequality of order k for 1 k n 1

an m estimator for tail dependence in arbitrary dimensions consider a random sample in the max domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model a new estimator for the unknown parameter is defined as the value that minimizes the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts the minimization problem has with probability tending to one a unique global solution the estimator is consistent and asymptotically normal the spectral measures of the tail dependence models to which the method applies can be discrete or continuous examples demonstrate the applicability and the performance of the method

symmetry classification and exact solutions of the kramers equation the complete classification of lie symmetries for the kramers equation is described group invariance under infinitesimal transformations is used to generate a wide class of exact solutions some of the known solutions are obtained as particular cases the general results can also be used to solve some types of moving boundary problems we researched some q conditional symmetry properties of this equation

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

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

non parametric bayesian inference on bivariate extremes the tail of a bivariate distribution function in the domain of attraction of a bivariate extreme value distribution may be approximated by the one of its extreme value attractor the extreme value attractor has margins that belong to a three parameter family and a dependence structure which is characterised by a probability measure on the unit interval with mean equal to one half called spectral measure inference is done in a bayesian framework using a censored likelihood approach a prior distribution is constructed on an in nite dimensional model for this measure the model being at the same time dense and computationally manageable a trans dimensional markov chain monte carlo algorithm is developed and convergence to the posterior distribution is established in simulations the bayes estimator for the spectral measure is shown to compare favorably with frequentist nonparametric estimators an application to a data set of danish re insurance claims is provided

equivalence and symmetries for variable coefficient linear heat type equations i a systematic and unified approach to transformations and symmetries of a class of variable coefficient heat type linear partial differential equations pdes is presented the complete symmetry group classification is re performed in the current context a useful criterion which is necessary and sufficient for being reducible to the standard heat equation by point transformations is established a similar criterion is also valid for the equations to have a four or six dimensional symmetry group nontrivial symmetry groups in this situation the basis elements are listed in terms of coefficients a number of illustrative examples are given in particular some applications from the recent literature are re examined in our new approach multidimensional parabolic pdes of heat and schrodinger type are also considered

quantum channels irreducibly covariant with respect to the finite group generated by the weyl operators in matrix algebras we introduce a class of linear maps that are irreducibly covariant with respect to the finite group generated by the weyl operators in particular we analyze the irreducibly covariant quantum channels that is the completely positive and trace preserving linear maps interestingly imposing additional symmetries leads to the so called generalized pauli channels which were recently considered in the context of the non markovian quantum evolution finally we provide examples of irreducibly covariant positive but not necessarily completely positive maps

estimating the probability of a rare event let x 1 y 1 x 2 y 2 x n y n be a random sample from a bivariate distribution function f which is in the domain of attraction of a bivariate extreme value distribution function g a subset c of r 2 is given which contains none of the observations we shall give an asymptotic confidence interval for pr x i y i c under certain conditions

generalized choi maps in three dimensional matrix algebra we consider a class of positive linear maps in the three dimensional matrix algebra which are generalizations of the positive linear map constructed by choi in the relation with positive semidefinite biquadratic forms we find conditions for which such maps are completely positive completely copositive decomposable and two positive

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

fisher information for a multivariate extreme value distribution summary explicit algebraic formulae for the fisher information matrix of the multivariate extreme value distribution with generalised extreme value margins and logistic dependence structure are given

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

symmetries of the fokker planck equation with a constant diffusion matrix in 2 1 dimensions we completely classify the symmetries of the fokker planck equation in two spatial dimensions with a constant positive definite diffusion matrix we apply these results to construct group invariant solutions for a physically interesting family of fokker planck equations

regional modelling of extreme storms via max stable processes asymptotic models for extremes of random processes often form the basis for estimating the extremal behaviour of environmental phenomena most such phenomena have a spatial dimension and the aim of this paper is to develop a procedure for modelling in continuous space the spatial dependence within extreme events a principal objective in the analysis as with other current research on extremes is to base inference on as much of the available data as possible the modelling procedures are justified on simulated data and subsequently applied to a series of rainfall data

generalising wigner s theorem we analyse linear maps of operator algebras mapping the set of rank k projectors onto the set of rank l projectors surjectively a complete characterisation of such maps for prime is provided a particular case corresponding to is well known as wigner s theorem hence our result may be considered as a generalisation of this celebrated wigner s result

lie symmetries of some equations of the fokker planck type the structure of the local lie groups of symmetries of some partial differential equations of the fokker planck type in one space dimension is investigated a connection between these groups and the group sl2 r is established in the sense that they are all shown to be locally isomorphic to sl2 r a where a is the radical it is conjectured that the groups of lie symmetries of all fokker planck equations in one space dimension have this structure the notion of partial invariance due to ovsiannikov is applied to the equations studied it appears plausible that the class of partially invariant solutions of these equations is larger than the class of invariant solutions although no explicit demonstration of this claim is available at present

positive maps from irreducibly covariant operators in this paper we discuss positive maps induced by irreducibly covariant linear operators for finite groups the application of group theory methods allows deriving some new results of a different kind in particular a family of necessary conditions for positivity for such objects is derived stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map in the low dimensional cases for the permutation group s 3 and the quaternion group q the necessary and sufficient conditions are given together with the discussion on their decomposability in higher dimensions we present positive maps induced by a three dimensional irreducible representation of the permutation group s 4 and d dimensional representation of the monomial unitary group mu d in the latter case we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction we show that the generalised choi map can be obtained by considered covariant maps induced by the monomial unitary group as an additional result a novel interpretation of the fujiwara algolet conditions for positivity and complete positivity is presented finally in the end a new form of an irreducible representation of the symmetric group s n is constructed allowing us to simplify the form of certain choi jamiolkowski images derived for irreducible representations of the symmetric group

bivariate extreme value distributions an application of the gibbs sampler to the analysis of floods the quantitative definition of the magnitude of a flood episode exclusively in terms of its peak rate of discharge is often insufficient in common with most episodic hydrological phenomena floods are intrinsically multivariate random events characterized not only by their peak flow but also by their volume and the duration of discharge above critical thresholds a complete definition in terms of these three component random variables has not found its way into standard analytical practice this may be partly because the theory of multivariate extremes is not straightforward and partly because no intuitively attractive way of presenting the component variables in a probabilistic framework has been put forward we propose that the gibbs sampler a member of the new generation of computer intensive methods in computational statistics bypasses the theoretical difficulties we demonstrate the gibbs sampler and provide graphical presentations of the results for the annual flood hydrograph on the mekong river at vientiane laos we extend the results to evaluate the potential distribution of flood damage to the city

ordered multivariate extremes multivariate extreme value models and associated statistical methods are developed for vector observations whose components are subject to an order restriction the approach extends the multivariate threshold methodology of coles and tawn joe and co workers and smith and co workers the results are illustrated by an analysis of extreme rainfalls of different durations and by a study of the problem of linking a long series of daily rainfall extremes with a partially overlapping shorter series of hourly extremes

spectral conditions for positive maps we provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions this construction generalizes the celebrated choi example of a map which is positive but not completely positive it is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements such maps provide basic tools to study quantum entanglement in multipartite systems

ovsiannikov s method and the construction of partially invariant solutions a plausibility argument presented by the first two authors in an earlier paper j math phys 26 3042 1985 concerning the existence of partially invariant solutions for some equations of the fokker planck type is made precise by the explicit construction of one such solution in the process a substantial simplification of ovsiannikov s method for finding partially invariant solutions is achieved in addition the class of partially invariant solutions obtained by ovsiannikov for the equations of transonic flow of a gas is enlarged

estimation of quantiles of the maximum of n observations summary given independent and identically distributed random variables from an unknown distribution f we want to estimate quantiles of fn using asymptotic extreme value theory two parametric approximations to fn are considered ahd approximate maximum likelihood estimates are obtained conditions for consistency and asymptotic normality of these estimates are given and properties of the estimates are obtained through a small simulation study

ozone concentrations a robust analysis of multivariate extremes data on ozone concentrations at four monitoring sites in central canada are examined univariate extreme value methods do not allow for the required inference and multivariate methods exploiting the joint dependence of the data are necessary families of multivariate extreme value copulas with a flexible dependence structure and closed form cumulative distribution functions have recently been derived this article shows that maximum likelihood estimates of these parametric models may be obtained numerically and that their use can lead to increased efficiency in the estimation of margins we do not however have any assurance that all data are well modeled by these distributions because the space of multivariate extreme value copulas is infinite dimensional and data on ozone levels are prone to outliers this article addresses the robust methods required for proper analysis

on positive linear maps between matrix algebras abstract there is a concrete example of a positive linear map from m 2 to m 4 which is not decomposable modification of this map gives an explicit counterexample to a conjecture of woronowicz on the strong kadison inequality

symmetry classification and exact solutions of the one dimensional fokker planck equation with arbitrary coefficients of drift and diffusion symmetry properties of the one dimensional fokker planck equations with arbitrary coefficients of drift and diffusion are investigated it is proved that the group symmetry of these equations can be one two four or six parametric and corresponding criteria are obtained the changes of the variables reducing fokker planck equations to the heat equation and schrodinger equation with certain potentials are determined using the substructure of the invariance algebra of the rayleigh process equation we obtain the differential invariants and construct some classes of the exact solutions of this equation

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 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

maximum empirical likelihood estimation of the spectral measure of an extreme value distribution consider a random sample from a bivariate distribution function f in the max domain of attraction of all extreme value distribution function g this g is characterized by two extreme value indices and a spectral measure the latter determining the tail dependence structure of f a major issue in multivariate extreme value theory is the estimation of the spectral measure 1 p with respect to the l p norm for every p is an element of 1 infinity a nonparametric maximum empirical likelihood estimator is proposed for phi p the main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized asymptotic normality of the estimators is proved under conditions that allow for tail independence moreover the conditions are easily verifiable as we demonstrate through a number of theoretical examples a simulation study shows a substantially improved performance of the new estimators two case studies illustrate how to implement the methods in practice

classification of the extended symmetries of fokker planck equations the authors obtain a complete classification of the possible types of the extended symmetry of any fokker planck equation together with a necessary and sufficient condition for each type they give also the most general form up to a change of variables of the lie generators of these symmetries and a comparison with the case of the heat equation

potential symmetries and invariant solutions for the generalized one dimensional fokker planck equation abstract pucci and saccomandi j phys a 26 1993 681 have obtained invariant solutions for the fokker planck fp equation via potential symmetries herein we analyze the generalized one dimensional fp equation for various cases corresponding to the physically interesting situations through the application of the same technique some interesting outcomes of this study are the deductions of the new exact solutions that do not seem to have been reported in the literature

models for the extremes of markov chains the modelling of extremes of a time series has progressed from the assumption of independent observations to more realistic forms of temporal dependence in this paper we focus on markov chains deriving a class of models for their joint tail which allows the degree of clustering of extremes to decrease at high levels overcoming a key limitation in current methodologies theoretical aspects of the model are examined and a simulation algorithm is developed through which the stochastic properties of summaries of the extremal txhaviour of the chain are evaluated the approach is illustrated through a simulation study of extremal events of gaussian autoregressive processes and an application to temperature data