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

inference for stereological extremes in the production of clean steels the occurrence of imperfections so called inclusions is unavoidable the strength of a clean steel block is largely dependent on the size of the largest imperfection that it contains so inference on extreme inclusion size forms an important part of quality control sampling is generally done by measuring imperfections on planar slices leading to an extreme value version of a standard stereological problem how to make inference on large inclusions using only the sliced observations under the assumption that inclusions are spherical this problem has been tackled previously using a combination of extreme value models stereological calculations a bayesian hierarchical model and standard markov chain monte carlo mcmc techniques our objectives in this article are twofold 1 to assess the robustness of such inferences with respect to the assumption of spherical inclusions and 2 to develop an inference procedure that is valid for nonspherical inclusions we inv

pairwise likelihood methods for inference in image models summary we consider parameter estimation for certain random set models for image data for the spatial models considered here a full likelihood based approach to this problem is often difficult since the likelihood generally cannot be computed for images of typical size this has motivated the consideration of various less computationally demanding parameter estimation techniques we discuss one method based on likelihoods for small subsets of the data consideration of the basic technique within the framework of the theory of estimating functions leads to a weighted version of the basic approach results of a simulation study are reported for some variants of the boolean model and excursion sets of random fields the methods are illustrated for some data on the spatial incidence of heather

stationary min stable stochastic processes we consider the class of stationary stochastic processes whose margins are jointly min stable we show how the scalar elements can be generated by a single realization of a standard homogeneous poisson process on the upper half strip 0 1 x r and a group of l1 isometries we include a dobrushin like result for the realizations in continuous time

inference for clustered data using the independence loglikelihood we use the properties of independence estimating equations to adjust the independence loglikelihood function in the presence of clustering the proposed adjustment relies on the robust sandwich estimator of the parameter covariance matrix which is easily calculated the methodology competes favourably with established techniques based on independence estimating equations we provide some insight as to why this is so the adjustment is applied to examples relating to the modelling of wind speed in europe and annual maximum temperatures in the u k

finite size scaling of the bayesian perceptron we study numerically the properties of the bayesian perceptron through a gradient descent on the optimal cost function the theoretical distribution of stabilities is deduced it predicts that the optimal generalizer lies close to the boundary of the space of error free solutions the numerical simulations are in good agreement with the theoretical distribution the extrapolation of the generalization error to infinite input space size agrees with the theoretical results finite size corrections are negative and exhibit two different scaling regimes depending on the training set size the variance of the generalization error vanishes for n confirming the property of self averaging s1063 651x 97 15406 x

quasi likelihood and its application a general approach to optimal parameter estimation the general framework an alternative approach e sufficiency asymptotic confidence zones of minimum size asymptotic quasi likelihood combining estimating functions projected quasi likelihood bypassing the likelihood hypothesis testing infinite dimensional problems miscellaneous applications consistency and asymptotic normality for estimating functions complements and strategies for application

modeling spatial extremes via ensemble of trees of pairwise copulas assessing the risk of extreme events in a spatial domain such as hurricanes floods and droughts presents a unique significance in practice unfortunately the existing extreme value statistical models are typically not feasible for practical large scale problems graphical models on the other hand are capable of handling sizable number of variables but have yet to be explored in the realm of extreme value analysis to bridge the gap an extreme value graphical model is introduced in this paper i e an ensemble of trees of pairwise copulas etpc in the proposed graphical model extreme value marginal distributions are stitched together by means of a pairwise copulas which in turn are the building blocks of the ensemble of trees novel linear complexity stochastic gradient based algorithms are then developed for learning the etpc model and inferring missing data as a result the etpc model is applicable to extreme value problems with thousands of variables it can be proven that under mild conditions the etpc model exhibits the favorable property of tail dependence between an arbitrary pair of sites variables consequently the model is able to reliably capture statistical dependence between extreme values at different sites experimental results for both synthetic and real data demonstrate the advantages of the etpc model in modeling fitting imputation and computational efficiency

a note on composite likelihood inference and model selection a composite likelihood consists of a combination of valid likelihood objects usually related to small subsets of data the merit of composite likelihood is to reduce the computational complexity so that it is possible to deal with large datasets and very complex models even when the use of standard likelihood or bayesian methods is not feasible in this paper we aim to suggest an integrated general approach to inference and model selection using composite likelihood methods in particular we introduce an information criterion for model selection based on composite likelihood we also describe applications to the modelling of time series of counts through dynamic generalised linear models and to the analysis of the well known old faithful geyser dataset

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

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

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

multivariate max stable spatial processes summary max stable processes allow the spatial dependence of extremes to be modelled and quantified so they are widely adopted in applications for a better understanding of extremes it may be useful to study several variables simultaneously to this end we study the maxima of independent replicates of multivariate processes both in the gaussian and student t cases we define a poisson process construction and introduce multivariate versions of the smith gaussian extreme value the schlather extremal gaussian and extremal t and the brown resnick models we develop inference for the models based on composite likelihoods we present results of monte carlo simulations and an application to daily maximum wind speed and wind gust

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

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

composite likelihood estimation for the brown resnick process summary genton et al 2011 investigated the gain in efficiency when triplewise rather than pairwise likelihood is used to fit the popular smith max stable model for spatial extremes we generalize their results to the brown resnick model and show that the efficiency gain is substantial only for very smooth processes which are generally unrealistic in applications

modelling extremes of the areal rainfall process risk assessment for many hydrological structures requires an estimate of the extremal behaviour of the rainfall regime within a specified catchment region in most cases it is the spatially aggregated rainfall which is the key process though in practice only pointwise rainfall measurements from a network of sites over the region are available in this paper we address the problem of making inferences about the extremal properties of the aggregated process from the pointwise data working within the usual extreme value paradigm a model is derived in which the resulting distribution is determined by the marginal tail behaviour and spatial dependence at extreme levels of the process data collected from a region in the south west of england are used to illustrate the procedure

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

on series representations of infinitely divisible random vectors general results on series representations involving arrival times in a poisson process are established for infinitely divisible banach space valued random vectors without gaussian components applying these results various generalization of lepage s representation are obtained in a unified way certain conditionally gaussian infinitely divisible random vectors are characterized and some problems related to a gaussian randomization method are investigated

a note on pseudolikelihood constructed from marginal densities for likelihood based inference involving distributions in which high dimensional dependencies are present it may be useful to use approximate likelihoods based for example on the univariate or bivariate marginal distributions the asymptotic properties of formal maximum likelihood estimators in such cases are outlined in particular applications in which only a single q x i vector of observations is observed are examined conditions under which consistent estimators of parameters result from the approximate likelihood using only pairwise joint distributions are studied some examples are analysed in detail

on the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max stable distributions full likelihood based inference for high dimensional multivariate extreme value distributions or max stable processes is feasible when incorporating occurrence times of the maxima without this information ddd dimensional likelihood inference is usually precluded due to the large number of terms in the likelihood however some studies have noted bias when performing high dimensional inference that incorporates such event information particularly when dependence is weak we elucidate this phenomenon showing that for unbiased inference in moderate dimensions dimension ddd should be of a magnitude smaller than the square root of the number of vectors over which one takes the componentwise maximum a bias reduction technique is suggested and illustrated on the extreme value logistic model

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

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

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

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

information gain and a general measure of correlation summary given a parametric model of dependence between two random quantities x and y the notion of information gain can be used to define a measure of correlation this definition of correlation generalizes both the usual product moment correlation coeffi cient for the bivariate normal model and the multiple correlation coefficient in the standard linear regression model the use of this information based correlation in a descriptive statistical analysis is examined and several examples are given if the dependence between two random quantities x and y is modelled parametri cally then the concept of information gain can be used to define a measure of correlation this correlation coefficient can appear in two possible contexts depending on whether one models the joint distribution of x and y or just the conditional distribution of y given x an important motivating feature for this information based correlation coefficient is the fact that it generalizes both the usual product moment correlation coefficient for the bivariate normal model and the usual multiple correlation coefficient for the standard multiple regression model with normal errors our intuition is well developed for these usual correlation coefficients and hopefully our intuition will still be applicable for the information based correlation in more general modelling situations of parametric dependence further since our correlation coefficient is based on information gain we might hope to extend our intuition to interpret the information gain in any statistical modelling situation where we want to assess how much better a more complicated model is than a simpler model the concept of information gain for general statistical models is described in 2 this concept is then used to define an information based measure of correlation the joint case is covered in 3 and the conditional case in 4 estimation of the correlation coefficient is carried out by estimating the corresponding information gain see 5 7 the use of information gain for the purpose of model choice in a descriptive statistical analysis is discussed in 8 and a comparison between our approach and akaike s information criterion is given in 9 some examples of the use of this information based correlation are given in 10 and 11

assessing the performance of the independence method in modeling spatial extreme rainfall spatial statistical methods are often employed to improve precision when estimating marginal distributions of extreme rainfall methods such as max stable and copula models parameterize the spatial dependence and provide a continuous spatial representation alternatively the independence method can be used to estimate marginal parameters without the need for parameterizing the spatial dependence and this method has been under utilized in hydrologic applications this paper investigates the effectiveness of the independence method for marginal parameter estimation of spatially dependent extremes its performance is compared with three spatial dependence models max stable brown resnick max stable schlather and gaussian copula by means of a simulation study the independence method is statistically robust in estimating parameters and their associated confidence intervals for spatial extremes with various underlying dependence structures the spatial dependence models perform comparably with the independence method when the spatial dependence structure is correctly specified otherwise they exhibit considerably worse performance we conclude that the independence method is more appealing for modeling the marginal distributions of spatial extremes e g regional estimation of trends in rainfall extremes due to its greater robustness and simplicity the four statistical methods are illustrated using a spatial data set comprising 69 subdaily rainfall series from the greater sydney region australia

statistical inference for max stable processes in space and time max stable processes have proved to be useful for the statistical modelling of spatial extremes several families of max stable random fields have been proposed in the literature one such representation is based on a limit of normalized and rescaled pointwise maxima of stationary gaussian processes that was first introduced by kabluchko and co workers this paper deals with statistical inference for max stable space time processes that are defined in an analogous fashion we describe pairwise likelihood estimation where the pairwise density of the process is used to estimate the model parameters for regular grid observations we prove strong consistency and asymptotic normality of the parameter estimates as the joint number of spatial locations and time points tends to furthermore we discuss extensions to irregularly spaced locations a simulation study shows that the method proposed works well for these models