On Prediction of Bivariate Extremes

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 Capéerà 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 Rootzén 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.

並列關鍵字

Bivariate extreme value models ； Prediction regions ； Comparative study ； Wind speed extremes

國際替代計量

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