1,720,982 research outputs found
Empirical Bayes inference for the block maxima method
Get PDFThe block maxima method is one of the most popular approaches for extreme value analysis with independent and identically distributed observations in the domain of attraction of an extreme value distribution. The lack of a rigorous study on the Bayesian inference in this context has limited its use for statistical analysis of extremes. In this paper we propose an empirical Bayes procedure for inference on the block maxima law and its related quantities.We show that the posterior distributions of the tail index of the data distribution and of the return levels (representative of future extreme episodes) are consistent and asymptotically normal. These properties guarantee the reliability of posterior-based inference. We also establish contraction rates of the posterior predictive distribution, the key tool in Bayesian probabilistic forecasting. Posterior computations are readily obtained via an efficient adaptive Metropolis-Hasting type of algorithm. Simulations show its excellent inferential performances already with modest sample sizes. The utility of our proposal is showcased analysing extreme winds generated by hurricanes in Southeastern US
Consistency of Bayesian inference for multivariate max-stable distributions
No full textPredicting extreme events is important in many applications in risk analysis. Extreme-value theory suggests modelling extremes by max-stable distributions. The Bayesian approach provides a natural framework for statistical prediction. Although various Bayesian inferential procedures have been proposed in the literature of univariate extremes and some for multivariate extremes, the study of their asymptotic properties has been left largely untouched. In this paper we focus on a semiparametric Bayesian method for estimating max-stable distributions in arbitrary dimension. We establish consistency of the pertaining posterior distributions for fairly general, well-specified max-stable models, whose margins can be short-, light- or heavy-tailed. We then extend our consistency results to the case where data are samples of block maxima whose distribution is only approximately a max-stable one, which represents the most realistic inferential setting
Generalized Pareto copulas: a key to multivariate extremes
No full textThis paper discusses generalized Pareto copulas which are a key to multivariate extreme value theory. Any generalized Pareto copula can be represented in an easy analytical way using a particular type of D-norm. The characteristic property of a generalized Pareto copula is its exceedance stability
Strong Convergence of Peaks Over a Threshold
Get PDFExtreme Value Theory plays an important role to provide approximation results
for the extremes of a sequence of independent random variables when their
distribution is unknown. An important one is given by the {generalised Pareto
distribution} as an approximation of the distribution
of the excesses over a threshold , where is a suitable
norming function. In this paper we study the rate of convergence of
to in variational and Hellinger distances and
translate it into that regarding the Kullback-Leibler divergence between the
respective densities
Optimal weighted pooling for inference about the tail index and extreme quantiles
Get PDFThis paper investigates pooling strategies for tail index and extreme quantile estimation from heavy-tailed data. To fully exploit the information contained in several samples, we present general weighted pooled Hill estimators of the tail index and weighted pooled Weissman estimators of extreme quantiles calculated through a nonstandard geometric averaging scheme. We develop their large-sample asymptotic theory across a fixed number of samples, covering the general framework of heterogeneous sample sizes with different and asymptotically dependent
distributions. Our results include optimal choices of pooling weights based on asymptotic variance and MSE minimization. In the important application of distributed inference, we prove that the variance-optimal distributed estimators are asymptotically equivalent to the benchmark Hill and Weissman estimators based on the unfeasible combination of subsamples, while the AMSE-optimal distributed estimators enjoy a smaller AMSE than the benchmarks in the case of large bias. We consider additional scenarios where the number of subsamples grows with the total sample size and effective subsample sizes can be low. We extend our methodology to handle serial
dependence and the presence of covariates. Simulations confirm the statistical inferential theory of our pooled estimators. Two applications to real weather and insurance data are showcased
Estimation and uncertainty quantification for extreme quantile regions
No full textEstimation of extreme quantile regions, spaces in which future extreme events can occur with a given low probability, even beyond the range of the observed data, is an important task in the analysis of extremes. Existing methods to estimate such regions are available, but do not provide any measures of estimation uncertainty. We develop univariate and bivariate schemes for estimating extreme quantile regions under the Bayesian paradigm that outperforms existing approaches and provides natural measures of quantile region estimate uncertainty. We examine the method’s performance in controlled simulation studies. We illustrate the applicability of the proposed method by analysing high bivariate quantiles for pairs of pollutants, conditionally on different temperature gradations, recorded in Milan, Italy
Records for time-dependent stationary Gaussian sequences
No full textFor a zero-mean, unit-variance stationary univariate Gaussian process we derive the probability that a record at the time , say , takes place and derive its distribution function. We study the joint distribution of the arrival time process of records and the distribution of the increments between records. We compute the expected number of records. We also consider two consecutive and non-consecutive records, one at time and one at time and we derive the probability that the joint records occur as well as their distribution function. The probability that the records and take place and the arrival time of the -th record, are independent of the marginal distribution function, provided that it is continuous. These results actually hold for a strictly stationary process with Gaussian copulas
Tail Risk Inference via Expectiles in Heavy-Tailed Time Series
No full textExpectiles define the only law-invariant, coherent and elicitable risk measure apart from the expectation. The popularity of expectile-based risk measures is steadily growing and their properties have been studied for independent data, but further results are needed to establish that extreme expectiles can be applied with the kind of dependent time series models relevant to finance. In this article we provide a basis for inference on extreme expectiles and expectile-based marginal expected shortfall in a general beta-mixing context that encompasses ARMA and GARCH models with heavy-tailed innovations. Our methods allow the estimation of marginal (pertaining to the stationary distribution) and dynamic (conditional on the past) extreme expectile-based risk measures. Simulations and applications to financial returns show that the new estimators and confidence intervals greatly improve on existing ones when the data are dependent.STA
Comment on Article by Page and Quintana
No full textPage and Quintana (2016) introduce the novel methodology of spatial product partition models in order to explicitly model the partitioning of spatial locations, with the aim of balancing local and global spatial dependence. Here we first discuss Gibbs-type partitions and their connection to exchangeable product partition models and their possible use as building blocks of spatial product partition models. Then, adopting the viewpoint of extreme value theory, we focus on two approaches for modeling spatial extremes, namely hierarchical modeling based on a latent stochastic process and modeling based on max-stable processes. Additional insights and interesting findings may arise by developing the approach of Page and Quintana (2016) along these lines
Strong convergence of multivariate maxima
No full textIt is well known and readily seen that the maximum of n independent and uniformly on [0; 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalized Pareto copula. Sklar's theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima
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