Laurens de Haan

We consider extreme value analysis for independent but non-identically distributed observations. In particular, the observations do not share the same extreme value index. Assuming continuously changing extreme value indices, we provide a non-parametric estimate for the functional extreme value index. Besides estimating the extreme value index locally, we also provide a global estimator for the trend and its joint asymptotic theory. The asymptotic theory for the global estimator can be used for testing a pre-specified parametric trend in the extreme value indices. In particular, it can be applied to test whether the extreme value index remains at a constant level across all observations.

In environmental applications of extreme value statistics, the underlying stochastic process is often modeled either as a max-stable process in continuous time/space or as a process in the domain of attraction of such a max-stable process. In practice, however, the processes are typically only observed at discrete points and one has to resort to interpolation to fill in the gaps. We discuss the influence of such an interpolation on estimators of marginal parameters as well as estimators of the exponent measure. In particular, natural conditions on the fineness of the observational scheme are developed which ensure that asymptotically the interpolated estimators behave in the same way as the estimators which use fully observed continuous processes.

A new approach for evaluating time-trends in extreme values accounting also for spatial dependence is proposed. Based on exceedances over a space-time threshold, estimators for a trend function and for extreme value parameters are given, leading to a homogenization procedure for then applying stationary extreme value processes. Extremal dependence over space is further evaluated through variogram analysis including anisotropy. We detect significant inhomogeneities and trends in the extremal behaviour of daily precipitation data over a time period of 84 years and from 68 observational weather stations in North-West Germany. We observe that the trend is not monotonous over time in general. Asymptotic normality of the estimators under maximum domain of attraction conditions are proven.

Project Euclid - mathematics and statistics online

The estimation of the extremal dependence structure is spoiled by the impact of the bias, which increases with the number of observations used for the estimation. Already known in the univariate setting, the bias correction procedure is studied in this paper under the multivariate framework. New families of estimators of the stable tail dependence function are obtained. They are asymptotically unbiased versions of the empirical estimator introduced by Huang [Statistics of bivariate extremes (1992) Erasmus Univ.]. Since the new estimators have a regular behavior with respect to the number of observations, it is possible to deduce aggregated versions so that the choice of the threshold is substantially simplified. An extensive simulation study is provided as well as an application on real data.

In extreme value theory, there are two fundamental approaches, both widely used: the block maxima (BM) method and the peaks-over-threshold (POT) method. Whereas much theoretical research has gone into the POT method, the BM method has not been studied thoroughly. The present paper aims at providing conditions under which the BM method can be justified. We also provide a theoretical comparative study of the methods, which is in general consistent with the vast literature on comparing the methods all based on simulated data and fully parametric models. The results indicate that the BM method is a rather efficient method under usual practical conditions. In this paper, we restrict attention to the i.i.d. case and focus on the probability weighted moment (PWM) estimators of Hosking, Wallis and Wood [Technometrics (1985) 27 251–261].

Summary: We extend classical extreme value theory to non-identically distributed observations. When the tails of the distribution are proportional much of extreme value statistics remains valid. The proportionality function for the tails can be estimated non-parametrically along with the (common) extreme value index. For a positive extreme value index, joint asymptotic normality of both estimators is shown; they are asymptotically independent. We also establish asymptotic normality of a forecasted high quantile and develop tests for the proportionality function and for the validity of the model. We show through simulations the good performance of the procedures and also present an application to stock market returns. A main tool is the weak convergence of a weighted sequential tail empirical process.