Bivariate extreme value distributions can be used to model dependence of observations from random variables in extreme levels. There is no finite dimensional parametric family for these distributions, but they can be characterized by a certain one-dimensional function which is known as Pickands dependence function. In many applications the general approach is to estimate the dependence function with a non-parametric method and then conduct further analysis based on the estimate. Although this approach is flexible in the sense that it does not impose any special structure on the dependence function, its main drawback is that the estimate is not available in a closed form. This paper provides some theoretical results which can be used to find a closed form approximation for an exact or an estimate of a twice differentiable dependence function and its derivatives. We demonstrate the methodology by calculating approximations for symmetric and asymmetric logistic dependence functions and their second derivatives. We show that the theory can be even applied to approximating a non-parametric estimate of dependence function using a convex optimization algorithm. Other discussed applications include a procedure for testing whether an estimate of dependence function can be assumed to be symmetric and estimation of the concordance measures of a bivariate extreme value distribution. Finally, an Australian annual maximum temperature dataset is used to illustrate how the theory can be used to build semi-infinite and compact predictions regions.

Summary We suggest two methods for simulating from a multivariate copula in an arbitrary dimension. Although our main emphasis in this paper is on multivariate extreme value distributions, the proposed methods can be applied to any copula. The basic idea is to approximate the (unknown) density of the copula by a distribution that has a piece‐wise constant (histogram) density. This is achieved by partitioning the support of a given copula C into a large number of hyper‐rectangles and using them to generate random variates from an approximation of the copula. We suggest two methods for finding this approximation which correspond to either finding hyper‐rectangles which have equal probability mass with respect to C , or determining a partition using hyper‐squares of equal volume and finding the corresponding probability mass of each hyper‐square. We also discuss how the generated random variates can be used as proposals in a Metropolis–Hastings algorithm, when C is an absolutely continuous distribution function, to generate a sequence of random variates from C . An implementation of the proposed methodologies is provided for the statistical computing and graphics environment R in our package called SimCop .

We propose a methodology based on multivariate extreme value theory, to analyze the dependence between markets during the nancial crisis. We argue that extreme dependence based on block maximum is a more appropriate measure to study dependence between stock markets, when a regime shifts, than other alternatives. With this methodology, we are able to detect the increase in the extreme dependences between US and other markets during the 2008 financial crisis. In addition, the estimated dependent function allows one to quantify maximum impact of the crisis on each individual market. This enables us to conduct a more constructive stress test analysis. Finally, the approach can be extended to gain further insight into the contagion literature.

There are mainly two competing approaches to modeling high dimensional extremes, namely multivariate extreme value distributions and multivariate peaks over threshold models which lead to a class of distribution called multivariate generalized Pareto distributions. Although the probability theory for the latter models is fairly well developed the statistical properties of them are generally unknown. We compare performances of these models for prediction of extremes under different circumstances and apply the results to modeling of real wind speed data. When modeling such extreme events one should not leave out of consideration that observations measured in closely located stations usually show strong dependence. Thus, besides fitting univariate margins, the knowledge of the dependence structure among the stations is also crucial. For the bivariate maxima the parametric cases are fully developed, but these structures are not always fiexible enough for real applications. A promising alternative way f or modeling the dependence could be obtained by non-parametric dependence functions. The most efficient known non-parametric models were introduced by Capeera et al. [3] and Hall and Tajvidi [7]. However to obtain density estimation further refinements are needed, since these approaches do not result in dependence functions which are differentiable everywhere. To tackle this problem polynomial smoothing splines have been considered taking into account all required constraints on dependence functions. It should be noted that investigating “only” the maxima can hide the time structure within the given period, so we do not know whether the different components of the maxima occurred really simultaneously (e.g., in the same day) or not. To avoid this problem exceedances over a high threshold can be considered. We applied a new definition for describing the distribution of the exceedances proposed by Rootzen and Tajvidi [11]. The main curiosity of it is including also those observations in modeling which are above the threshold at least in one component. Both of the approaches for maxima and exceedances have been applied for bivariate datasets arising from wind time series of the last 5 decades measured in north Germany. We compute prediction regions for all fitted models, which makes the models easily comparable. Finally, as the statistical properties of the proposed exceedance model is still not fully studied, we investigate its accuracy and compare it with rather standard block maxima approach by a simulation study.

Prediction from time-series data is traditionally accomplished using parametric, or at least structural, methods. For example, after removing trends, arguing that the time-series is an autoregression, and estimating the autoregressive parameters, we may predict future expected values, conditional on the past. In this paper, motivated by long meteorological maximum-temperature time-series, we suggest alternative approaches founded on functional data analysis. The new techniques make relatively few assumptions about the nature of the data, and allow consistent inference in cases where conventional models are inappropriate. They have both parametric and nonparametric forms. In the former context, our techniques are based on dimension-reduction and least-squares arguments; in the latter, they are founded on distance-based methods and statistical smoothing. We illustrate our method by application to Australian meteorological data, and by a simulation study designed to reflect those data. Theoretical arguments are used to demonstrate statistical consistency. (Less)

Consider a process of events on a line L, where, for the most part, the events occur randomly in both time and location. A scatterplot of the pair that represents position on the line, and occurrence time, will resemble a bivariate stochastic point process in a plane, P say. If, however, some of the points on L arise through a more regular phenomenon which travels along the line at an approximately constant speed, creating new points as it goes, then the corresponding points in P will occur roughly in a straight line. It is of interest to locate such lines, and thereby identify, as nearly as possible, the points on L which are associated with the (approximately) constant-velocity process. Such a problem arises in connection with the study of seismic data, where L represents a fault-line and the constant-velocity process there results from the steady diffusion of stress. We suggest methodology for solving this needle-in-a-haystack problem, and discuss its properties. The technique is applied to both simulated and real data. In the latter case it draws particular attention to events occurring along the San Andreas fault, in the vicinity of Parkville, California, on 5 April 1995.

Statistical inference for extremes has been a subject of intensive research over the past couple of decades. One approach is based on modelling exceedances of a random variable over a high threshold with the generalized Pareto (GP) distribution. This has proved to be an important way to apply extreme value theory in practice and is widely used. We introduce a multivariate analogue of the GP distribution and show that it is characterized by each of following two properties: first, exceedances asymptotically have a multivariate GP distribution if and only if maxima asymptotically are extreme value distributed; and second, the multivariate GP distribution is the only one which is preserved under change of exceedance levels. We also discuss a bivariate example and lower-dimensional marginal distributions.

Newly available wavelet bases on multi-resolution analysis have exciting implications for detection of change-points. By checking the absolute value of wavelet coefficients one call detect discontinuities in ail otherwise smooth curve even in the presence of additive noise. In this paper, we combine wavelet methods and extreme value theory to test the presence of ail arbitrary number of discontinuities in an unknown function observed with noise. Our approach is based on a Peaks Over Threshold modelling of noisy wavelet transforms. Particular features of our method include the estimation of the extreme value index in the tail of the noise distribution. The critical region of our test is, derived using a Generalised Pareto Distribution approximation to normalised sums. Asymptotic results show that our method is powerful in a wide range of medium size wavelet frequencies. We compare our test with competing approaches on simulated examples and illustrate the method on Dow-Jones data.

This paper suggests using a mixture of parametric and non-parametric methods to construct prediction regions in bivariate extreme-value problems. The non-parametric part of the technique is used to estimate the dependence function, or copula, and the parametric part is employed to estimate the marginal distributions. A bootstrap calibration argument is suggested for reducing coverage error. This combined approach is compared with a more parametric one, relative to which it has the advantages of being more flexible and simpler to implement. It also enjoys these features relative to predictive likelihood methods. The paper shows how to construct both compact and semi-infinite bivariate prediction regions, and it treats the problem of predicting the value of one component conditional on the other. The methods are illustrated by application to Australian annual maximum temperature data.

The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample sizes. By applying a general method of D. N. Lawley, correction factors for likelihood ratio statistics of parameters and quantiles of the GPD have been calculated. Simulations show that for small sample sizes accuracy of confidence intervals can be improved by incorporating the computed correction factors to the likelihood-based confidence intervals. While the modified likelihood method has better empirical coverage probability, the mean length of produced intervals are not longer than corresponding bootstrap confidence intervals. This article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.

Motivated by applications in high-dimensional settings, we suggest a test of the hypothesis H-0 that two sampled distributions are identical. It is assumed that two independent datasets are drawn from the respective populations, which may be very general. In particular, the distributions may be multivariate or infinite-dimensional, in the latter case representing, for example, the distributions of random functions from one Euclidean space to another. Our test uses a measure of distance between data. This measure should be symmetric but need not satisfy the triangle inequality, so it is not essential that it be a metric. The test is based on ranking the pooled dataset, with respect to the distance and relative to any fixed data value, and repeating this operation for each fixed datum. A permutation argument enables a critical point to be chosen such that the test has concisely known significance level, conditional on the set of all pairwise distances.

A feature that distinguishes extreme-value contexts from more conventional statistical problems is that in the former we often wish to make predictions well beyond the range of the data. For example, one might have a 10-year sequence of observations of a phenomenon, and wish to make forecasts for the next 20 to 30 years. It is generally unclear how such long ranges of extrapolation affect prediction. In the present paper, and for extremes from a distribution with regularly varying tails at infinity, we address this problem. We approach it in two ways: first, from the viewpoint of predictive inference under a model that is admittedly only approximate, and where the errors of greatest concern are caused by the interaction of long-range extrapolation with model misspecification; second, where the model is accurate but errors arise from a combination of extrapolation and the fact that the method is only approximate. In both settings we show that, in a way which can be defined theoretically and confirmed numerically, one can make predictions exponentially far into the future without committing serious errors.

This paper contains a study of the extent to which aggregate losses due to severe wind storms can be explained by wind measurements. The analysis is based on 12 years of data for a region, Ska § ne, in southern Sweden. A previous investigation indicated that wind measurements from six recording stations in Ska § ne was insufficient to obtain accurate prediction. The present study instead uses geostrophic winds calculated from pressure readings, at a regular grid of size 50 kilometres over Ska § ne. However, also this meteorological data set is seen to be insufficient for accurate prediction of insurance risk. The results indicate that currently popular methods of evaluating wind storm risks from meteorological data should not be used uncritically by insurers or reinsurers. Nevertheless, wind data does contain some information on insurance. risks. There is a need for further research on how to use this information to improve risk assessment.

A topic of major current interest in extreme-value analysis is the investigation of temporal trends. For example, the potential influence of "greenhouse" effects may result in severe storms becoming gradually more frequent, or in maximum temperatures gradually increasing, with time. One approach to evaluating these possibilities is to fit, to data, a parametric model for temporal parameter variation, as well as a model describing the marginal distribution of data at any given point in time. However, structural trend models can be difficult to formulate in many circumstances, owing to the complex way in which different factors combine to influence data in the form of extremes. Moreover, it is not advisable to fit trend models without empirical evidence of their suitability. In this paper, motivated by datasets on windstorm severity and maximum temperature, we suggest a nonparametric approach to estimating temporal trends when fitting parametric models to extreme values from a weakly dependent time series. We illustrate the method through applications to time series where the marginal distributions are approximately Pareto, generalized-Pareto, extreme-value or Gaussian. We introduce time-varying probability plots to assess goodness of fit, we discuss local-likelihood approaches to fitting the marginal model within a window and we propose temporal cross-validation for selecting window width. In cases where both location and scale are estimated together, the Gaussian distribution is shown to have special features that permit it to play a universal role as a "nominal" model for the marginal distribution.

Two new methods are suggested for estimating the dependence function of a bivariate extreme-value distribution. One is based on a multiplicative modification of an earlier technique proposed by Pickands, and the other employs spline smoothing under constraints. Both produce estimators that satisfy all the conditions that define a dependence function, including convexity and the restriction that its curve lie within a certain triangular region. The first approach does not require selection of smoothing parameters; the second does, and for that purpose we suggest explicit tuning methods, one of them based on cross-validation.

We argue that prediction intervals based on predictive likelihood do not correct for curvature with respect to the parameter value when they implicitly approximate an unknown probability density. Partly as a result of this difficulty, the order of coverage error associated with predictive intervals and predictive limits is equal to only the inverse of sample size. In this respect those methods do not improve on the simpler,'naive' or 'estimative' approach. Moreover, in cases of practical importance the latter can be preferable, in terms of both the size and sign of coverage error. We show that bootstrap calibration of both naive and predictive-likelihood approaches increases coverage accuracy of prediction intervals by an order of magnitude, and, in the case of naive intervals, preserves that method's numerical and analytical simplicity. Therefore, we argue, the bootstrap-calibrated naive approach is a particularly competitive alternative to more conventional, but more complex, techniques based on predictive likelihood.

Statistical extreme value theory provides a flexible and theoretically well motivated approach to the study of large losses in insurance. We give a brief review of the modem version of this theory and a “step by step” example of how to use it in large claims insurance. The discussion is based on a detailed investigation of a wind storm insurance problem. New results include a simulation study of estimators in the peaks over thresholds method with Generalised Pareto excesses, a discussion of Pareto and lognormal modelling and of methods to detect trends. Further results concern the use of meteorological information in the wind storm insurance and, of course, the results of the study of the wind storm claims.

In this paper we discuss a general approach to design and implementation of statistical computations. We use three previous articles as examples to illustrate diiculties which arise in this kind of application and methods which may be used to solve them. A common theme in these articles is univariate and multivariate generalised Pareto distributions. However, the discussed problems are of a rather general nature and demonstrate some typical tasks in applied statistical research. The rst one concerns maximum likelihood estimation of parameters in a rather complicated bivariate generalised Pareto distribution. In this application, the main emphasis is on decomposing the optimisation problem into a few stages and using diierent tools at each stage of the problem. Object oriented programming is a rather recent approach to program development, which we found quite useful. We next describe and comment on using S-plus as a main environment for a simulation study concerning conndence intervals and accuracy estimation for the generalised Pareto distribution. We design a number of objects and explain their responsibilities in the simulation. The implementation of the proposed design in S-plus is also presented. The last part of the paper is devoted to a general discussion of diierent stages in data analysis. We suggest some tools which can be used at each stage of the analysis.