impact of dependence in river flow data on flood frequency analysis based on regression in quantile plots analysis and solutions 1 quantile quantile plots qq plots are often applied to investigate tail behavior in extreme value analysis when used for flood frequency analysis based on river flow data dependence in these data affects the extreme value distribution s tail behavior in the qq plot this problem is investigated on the basis of both synthetic and observed river flow series the synthetic series are generated on the basis of a stochastic simulation procedure which allows the real statistical flow properties to be known through simulation study we demonstrate that serial dependence causes bending behavior in the qq plot in the region of higher quantiles this leads to the estimating error when the slope of the qq plot is used as an estimator in the extreme value analysis a correction method is proposed by modifying the plotting position using the extremal index

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

generalized additive models for exceedances of high thresholds with an application to return level estimation for u s wind gusts generalized additive model gam forms offer a flexible approach to capturing marginal variation such forms are used here to represent distributional variation in extreme values and presented in t

some characterizations of graphs by star complements abstract let be an eigenvalue of the graph g with multiplicity k a star complement for in g is an induced subgraph h g x such that x k and is not an eigenvalue of g x various graphs related to generalized line graphs or their complements are characterized by star complements corresponding to eigenvalues 2 or 1

on the exceedance point process for a stationary sequence it is known that the exceedance points of a high level by a stationary sequence are asymptotically poisson as the level increases under appropriate long range and local dependence conditions when the local dependence conditions are relaxed clustering of exceedances may occur based on poisson positions for the clusters in this paper a detailed analysis of the exceedance point process is given and shows that under wide conditions any limiting point process for exceedances is necessarily compound poisson more generally the possible random measure limits for normalized exceedance point processes are characterized sufficient conditions are also given for the existence of a point process limit the limiting distributions of extreme order statistics are derived as corollaries

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

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

estimating the extremal index summary the extremal index is an important parameter measuring the degree of clustering of process the extremal index a parameter in the interval 0 1 is the reciprocal of the mean cluster size apart from being of interest in its own right it is a crucial parameter for determining the limiting distribution of extreme values from the process in this paper we review current work on statistical estimation of the extremal index and consider an optimality criterion based on a bias variance trade off theoretical results are developed for a simple doubly stochastic process and it is argued that the main formula obtained is valid for a much wider class of processes the practical implications are examined through simulations and a real data example

time delay estimation with unknown spatially correlated gaussian noise the problem of estimating the difference in arrival times of a non gaussian signal at two spatially separated sensors is considered the signal is assumed to be corrupted by spatially correlated gaussian noises of unknown cross correlation the author proposes and analyzes two new classes of methods for time delay estimation based on higher order statistics the proposed methods are conceptually very similar to the traditional cross correlation based techniques in that the proposed criteria peak at a lag value equalizing true delay since the proposed methods are based upon higher order cumulant statistics of the data they result in estimators that remain unbiased in the presence of gaussian noise

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

star complements and switching in graphs star complements are used to construct switching equivalent graphs which share an eigenvalue of the same multiplicity a variant of the construction is used to obtain a characterization of the mclaughlin graph

time delay estimation in unknown gaussian spatially correlated noise a novel class of methods that estimate the difference in arrival time between signals corrupted by spatially correlated gaussian noise sources of unknown cross correlation is presented the methods are based on the idea of comparing the similarities between the two sensor measurements in higher order spectrum domains bispectrum rather than in the cross correlation domain it is demonstrated that estimation techniques based on higher order cumulants suppress the effect of correlated gaussian noise sources and therefore exhibit improved performance over generalized cross correlation methods results are reported for different types of signals lengths of data records and signal to noise ratios

a unified statistical model for hydrological variables including the selection of threshold for the peak over threshold method 1 this paper explores the use of a mixture model for determining the marginal distribution of hydrological variables consisting of a truncated central distribution that is representative of the central or main mass regime which for the cases studied is a lognormal distribution and of two generalized pareto distributions for the maximum and minimum regimes representing the upper and lower tails respectively the thresholds defining the limits between these regimes and the central regime are parameters of the model and are calculated together with the remaining parameters by maximum likelihood after testing the model with a simulation study we concluded that the upper threshold of the model can be used when applying the peak over threshold method this will yield an automatic and objective identification of the threshold presenting an alternative to existing methods the model was also applied to four hydrological data series two mean daily flow series the thames at kingston united kingdom and the guadalfeo river at orgiva spain and two daily precipitation series fort collins co usa and orgiva spain it was observed that the model improved the fit of the data series with respect to the fit obtained with the lognormal ln and in particular provided a good fit for the upper tail moreover we concluded that the proposed model is able to accommodate the entire range of values of some significant hydrological variables

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

inference for clusters of extreme values inference for clusters of extreme values of a time series typically requires the identification of independent clusters of exceedances over a high threshold the choice of declustering scheme often has a significant effect on estimates of cluster characteristics we propose an automatic declustering scheme that is justified by an asymptotic result for the times between threshold exceedances the scheme relies on the extremal index which we show may be estimated before declustering and supports a bootstrap procedure for assessing the variability of estimates

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

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

tail behaviour and extremes of two state markov switching autoregressive models we examine the tail behaviour and extremal cluster characteristics of two state markov switching autoregressive models where the first regime behaves like a random walk the second regime is a stationary autoregression and the generating noise is light tailed under additional technical conditions we prove that the stationary solution has asymptotically exponential tail and the extremal index is smaller than one the extremal index and the limiting cluster size distribution of the process are calculated explicitly for some noise distributions and simulated for others the practical relevance of the results is illustrated by examining extremal properties of a regime switching autoregressive process with gamma distributed noise already applied successfully in river flow modeling the limiting aggregate excess distribution is shown to possess weibull like tail in this special case

nonparametric bispectrum based time delay estimators for multiple sensor data when estimating time delays of signals from sensor outputs the bispectrum has advantages when the signal has a skewed distribution the authors provide nonparametric estimation approaches which exploit the two dimensional nature of the bispectrum when dealing with observations from more than a pair of sensors the new algorithms have computational advantages and in some cases show improved performance compared to the conventional nonparametric bispectrum based time delay estimation approach

the cumulant theory of cyclostationary time series ii development and applications for pt i see ibid vol 42 no 12 p 3387 3408 1994 the development of the theory of nonlinear processing of cyclostationary time series that is initiated in part i is continued a new type of cumulant for complex valued variables is introduced and used to generalize the temporal and spectral moments and cumulants for cyclostationary time series from real valued to complex valued time series the relations between the temporal and spectral moments and cumulants at the inputs and outputs of several signal processing operations are determined formulas for the temporal and spectral cumulants of complex valued pulse amplitude modulated time series are derived estimators for the temporal moments and cumulants and for the cyclic polyspectra are presented and their properties are discussed the performance of these estimators is illustrated by several computer simulation examples for pulse amplitude modulated time series the theory is applied to the problems of weak signal detection and interference tolerant time delay estimation

extreme values for stationary and markov sequences soit x n n 1 2 une suite strictement stationnaire de variables aleatoires a valeurs reelles soit m i j max x i 1 x j et m n m 0 n soit c n une suite de nombres reels on montre sous des circonstances generales que p m n c n p x 1 c n np mu 1 u u p u n cu n xu 1 cu n o pour toute suite pn satisfaisant certaines conditions de taux de croissance

peaks over thresholds modeling with multivariate generalized pareto distributions when assessing the impact of extreme events it is often not just a single component but the combined behavior of several components which is important statistical modeling using multivariate gen

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

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

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

extremes of markov chains with tail switching potential summary a recent advance in the utility of extreme value techniques has been the characteri zation of the extremal behaviour of markov chains this has enabled the application of extreme value models to series whose temporal dependence is markovian subject to a limitation that prevents switching between extremely high and extremely low levels for many applications this is sufficient but for others most notably in the field of finance it is common to find series in which successive values switch between high and low levels we term such series markov chains with tail switching potential and the scope of this paper is to generalize the previous theory to enable the characterization of the extremal properties of series displaying this type of behaviour in addition to theoretical developments a modelling procedure is proposed a simulation study is made to assess the utility of the model in inferring the extremal dependence structure of autoregressive conditional heteroscedastic processes which fall within the tail switching markov family and generalized autoregressive conditional heteroscedastic processes which do not being non markov in general finally the procedure is applied to model extremal aspects of a financial index extracted from the new york stock exchange compendium

star complements in regular graphs old and new results we survey results concerning star complements in finite regular graphs and note the connection with designs and strongly regular graphs in certain cases we include improved proofs along with new results on stars and windmills as star complements

extremal analysis of processes sampled at different frequencies the observed extremes of a discrete time process depend on the process itself and the sampling frequency we develop theoretical results which show how to account for the effect of sampling frequency on extreme values thus enabling us to analyse systematically extremal data from series with different sampling rates we present statistical methodology based on these results which we illustrate though simulations and by applications to sea waves and rainfall data

automatic declustering of rare events the analysis of events with low probability but disastrous impact entails understanding how they cluster in time we present an automatic three step procedure for identifying clusters estimating the cluster size distribution and constructing confidence intervals for the extremal index which measures the degree of clustering of rare events the third step combines empirical likelihood and parametric likelihood approaches simulations show that our new procedure performs very well for finite samples and outperforms previous methods in constructing confidence intervals for the extremal index when there is clustering in the data as well as in estimating probabilities for small clusters copyright 2013 oxford university press

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

inference for the limiting cluster size distribution of extreme values any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound poisson under appropriate long range dependence conditions typically exceedances appear in clusters the underlying poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes in the present paper we introduce estimators of the limiting cluster size probabilities which are constructed through a recursive algorithm we derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions we study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data

on graphs with complete bipartite star complements abstract let be an eigenvalue of the graph g with multiplicity m a star complement for in g is an induced subgraph g x such that x m and is not an eigenvalue of g x some general observations concerning graphs with the complete bipartite graph k r s r s 2 as a star complement are followed by a complete analysis of the case r 2 s 5 the results include a characterization of the schlafli graph and the construction of all the regular graphs which have k 2 5 as a star complement

co cliques and star complements in extremal strongly regular graphs abstract suppose that the positive integer is the eigenvalue of largest multiplicity in an extremal strongly regular graph g by interlacing the independence number of g is at most 4 2 4 2 star complements are used to show that if this bound is attained then either a 1 and g is the schlafli graph or b 2 and g is the mclaughlin graph

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

extremes and local dependence in stationary sequences extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction a a weak mixing condition restricting long range dependence b a local condition restricting the clustering of high level exceedances

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

the cumulant theory of cyclostationary time series i foundation the problem of characterizing the sine wave components in the output of a polynomial nonlinear system with a cyclostationary random time series input is investigated the concept of a pure nth order sine wave is introduced and it is shown that pure nth order sine wave strengths in the output time series are given by scaled fourier coefficients of the polyperiodic temporal cumulant of the input time series the higher order moments and cumulants of narrowband spectral components of time series are defined and then idealized to the case of infinitesimal bandwidth such spectral moments and cumulants are shown to be characterized by the fourier transforms of the temporal moments and cumulants of the time series it is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts to put the contributions of the paper in perspective a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided

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