Bayesian inference on extreme value distribution using upper record values

In this paper we address estimation and prediction problems for extreme value distributions under the assumption that the only available data are the record values. We provide some properties and pivotal quantities, and derive unbiased estimators for the location and rate parameters based on these properties and pivotal quantities. In addition, we discuss mean-squared errors of the proposed estimators and exact confidence intervals for the rate parameter. In Bayesian inference, we develop objective Bayesian analysis by deriving non informative priors such as the Jeffrey, reference, and probability matching priors for the location and rate parameters. We examine the validity of the proposed methods through Monte Carlo simulations for various record values of size and present a real data set for illustration purposes.     

A location-invariant non-positive moment-type estimator of the extreme value index

This paper investigates a class of location invariant non-positive moment-type estimators of extreme value index, which is highly flexible due to the tuning parameter involved. Its asymptotic expansions and its optimal sample fraction in terms of minimal asymptotic mean square error are derived. A small scale Monte Carlo simulation turns out that the new estimators, with a suitable choice of the tuning parameter driven by the data itself, perform well compared to the known ones. Finally, the proposed estimators with a bootstrap optimal sample fraction are applied to an environmental data set.     

Bayesian inference for generalized extreme value distributions via Hamiltonian Monte Carlo

In this article, we propose to evaluate and compare Markov chain Monte Carlo (MCMC) methods to estimate the parameters in a generalized extreme value model. We employed the Bayesian approach using traditional Metropolis-Hastings methods, Hamiltonian Monte Carlo (HMC), and Riemann manifold HMC (RMHMC) methods to obtain the approximations to the posterior marginal distributions of interest. Applications to real datasets and simulation studies provide evidence that the extra analytical work involved in Hamiltonian Monte Carlo algorithms is compensated by a more efficient exploration of the parameter space.     

Bias and size corrections in extreme value modeling

Extreme value theory models have found applications in myriad fields. Maximum likelihood (ML) is attractive for fitting the models because it is statistically efficient and flexible. However, in small samples, ML is biased to O(N^?1) and some classical hypothesis tests suffer from size distortions. This paper derives the analytical Cox–Snell bias correction for the generalized extreme value (GEV) model, and for the model's extension to multiple order statistics (GEVr). Using simulations, the paper compares this correction to bootstrap-based bias corrections, for the generalized Pareto, GEV, and GEVr. It then compares eight approaches to inference with respect to primary parameters and extreme quantiles, some including corrections. The Cox–Snell correction is not markedly superior to bootstrap-based correction. The likelihood ratio test appears most accurately sized. The methods are applied to the distribution of geomagnetic storms.     

First moment approximations for order statistics from the extreme value distribution

Approximate expressions of the first moment of the order statistics of standard extreme value distributions are proposed. We compare different previously given approximations with the exact values. The results show that the approximation given here fits the exact values better than previously given models.     

Skew generalized extreme value distribution: Probability-weighted moments estimation and application to block maxima procedure

ABSTRACT

Following the work of Azzalini (1985 Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Stat. 12:171–178.[Web of Science ®] , [Google Scholar] and 1986 Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistica 46:199–208. [Google Scholar]) on the skew-normal distribution, we propose an extension of the generalized extreme value (GEV) distribution, the SGEV. This new distribution allows for a better fit of maxima and can be interpreted as both the distribution of maxima when maxima are taken on dependent data and when maxima are taken over a random block size. We propose to estimate the parameters of the SGEV distribution via the probability-weighted moment method. A simulation study is presented to provide an application of the SGEV on block maxima procedure and return level estimation. The proposed method is also implemented on a real-life data.     

Exponentiated generalized Pareto distribution: Properties and applications towards extreme value theory

The GPD is a central distribution in modelling heavy tails in many applications. Applying the GPD to actual datasets however is not trivial. In this paper we propose the Exponentiated GPD (exGPD), created via log-transform of the GPD variable, which has less sample variability. Various distributional quantities of the exGPD are derived analytically. As an application we also propose a new plot based on the exGPD as an alternative to the Hill plot to identify the tail index of heavy tailed datasets, and carry out simulation studies to compare the two.     

A new test for the extreme value distribution

In this paper a test statistic which is a modification of the W statistic for testing the goodness of fit for the two paremeter extreme value (smallest element) distribution is proposed. The test statistic Is obtained as the ratio of two linear estimates of the scale parameter. It Is shown that the suggested statistic is computationally simple and has good power properties. Percentage points of the statistic are obtained by performing Monte Carlo experiments. An example is given to illustrate the test procedure.     

Minimum distance estimators in extreme value distributions

We define minimum distance estimators for the parameters of the extreme value distribution G_o based on the Cramer-von-Mises distance. These estimators are rather robust and consistent, but asymptotically less efficient than the maximum likelihood estimators which are not robust. A small simulation study for finite sample size show that under G_o the finite efficiency of the minimum distance estimators is rather similar to the maximum likelihood ones.     

Kernel estimators for the second order parameter in extreme value statistics

We develop and study in the framework of Pareto-type distributions a general class of kernel estimators for the second order parameter ρ$\rho$, a parameter related to the rate of convergence of a sequence of linearly normalized maximum values towards its limit. Inspired by the kernel goodness-of-fit statistics introduced in Goegebeur et al. (2008), for which the mean of the normal limiting distribution is a function of ρ$\rho$, we construct estimators for ρ$\rho$ using ratios of ratios of differences of such goodness-of-fit statistics, involving different kernel functions as well as power transformations. The consistency of this class of ρ$\rho$ estimators is established under some mild regularity conditions on the kernel function, a second order condition on the tail function 1−F of the underlying model, and for suitably chosen intermediate order statistics. Asymptotic normality is achieved under a further condition on the tail function, the so-called third order condition. Two specific examples of kernel statistics are studied in greater depth, and their asymptotic behavior illustrated numerically. The finite sample properties are examined by means of a simulation study.     

A semi-parametric model for multivariate extreme values

Threshold methods for multivariate extreme values are based on the use of asymptotically justified approximations of both the marginal distributions and the dependence structure in the joint tail. Models derived from these approximations are fitted to a region of the observed joint tail which is determined by suitably chosen high thresholds. A drawback of the existing methods is the necessity for the same thresholds to be taken for the convergence of both marginal and dependence aspects, which can result in inefficient estimation. In this paper an extension of the existing models, which removes this constraint, is proposed. The resulting model is semi-parametric and requires computationally intensive techniques for likelihood evaluation. The methods are illustrated using a coastal engineering application.     

Modeling rare events through a pRARMAX process

7 and 8 introduce a power max-autoregressive process, in short pARMAX, as an alternative to heavy tailed ARMA when modeling rare events. In this paper, an extension of pARMAX is considered, by including a random component which makes the model more applicable to real data. We will see conditions under which this new model, here denoted as pRARMAX, has unique stationary distribution and we analyze its extremal behavior. Based on Bortot and Tawn (1998), we derive a threshold-dependent extremal index which is a functional of the coefficient of tail dependence of 14 and 15 which in turn relates with the pRARMAX parameter. In order to fit a pRARMAX model to an observed data series, we present a methodology based on minimizing the Bayes risk in classification theory and analyze this procedure through a simulation study. We illustrate with an application to financial data.     

A hierarchical model for extreme wind speeds

Summary.  A typical extreme value analysis is often carried out on the basis of simplistic inferential procedures, though the data being analysed may be structurally complex. Here we develop a hierarchical model for hourly gust maximum wind speed data, which attempts to identify site and seasonal effects for the marginal densities of hourly maxima, as well as for the serial dependence at each location. A Gaussian model for the random effects exploits the meteorological structure in the data, enabling increased precision for inferences at individual sites and in individual seasons. The Bayesian framework that is adopted is also exploited to obtain predictive return level estimates at each site, which incorporate uncertainty due to model estimation, as well as the randomness that is inherent in the processes that are involved.     

An extended Gaussian max-stable process model for spatial extremes

The extremes of environmental processes are often of interest due to the damage that can be caused by extreme levels of the processes. These processes are often spatial in nature and modelling the extremes jointly at many locations can be important. In this paper, an extension of the Gaussian max-stable process is developed, enabling data from a number of locations to be modelled under a more flexible framework than in previous applications. The model is applied to annual maximum rainfall data from five sites in South-West England. For estimation we employ a pairwise likelihood within a Bayesian analysis, incorporating informative prior information.     

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.     

Data-driven smooth tests for the extreme value distribution

In this paper we present data-driven smooth tests for the extreme value distribution. These tests are based on a general idea of construction of data-driven smooth tests for composite hypotheses introduced by Inglot, T., Kallenberg, W. C. M. and Ledwina, T. [(1997). Data-driven smooth tests for composite hypotheses. Ann. Statist., 25, 1222–1250] and its modification for location-scale family proposed in Janic-Wróblewska, A. [(2004). Data-driven smooth test for a location-scale family. Statistics, in press]. Results of power simulations show that the newly introduced test performs very well for a wide range of alternatives and is competitive with other commonly used tests for the extreme value distribution.     

Modelling extreme wind speeds in regions prone to hurricanes

Extreme wind speeds can arise as the result of a simple pressure differential, or a complex dynamic system such as a tropical storm. When sets of record values comprise a mixture of two or more different types of event, the standard models for extremes based on a single limiting distribution are not applicable. We develop a mixture model for extreme winds arising from two distinct processes. Working with sequences of annual maximum speeds obtained at hurricane prone locations in the USA, we take a Bayesian approach to inference, which allows the incorporation of prior information obtained from other sites. We model the extremal behaviour for the contrasting wind climates of Boston and Key West, and show that the standard models can give misleading results at such locations.     

Point and confidence interval estimates for a global maximum via extreme value theory

The aim of this paper is to provide some practical aspects of point and interval estimates of the global maximum of a function using extreme value theory. Consider a real-valued function f:D→? defined on a bounded interval D such that f is either not known analytically or is known analytically but has rather a complicated analytic form. We assume that f possesses a global maximum attained, say, at u*∈D with maximal value x*=max _u  f(u)?f(u*). The problem of seeking the optimum of a function which is more or less unknown to the observer has resulted in the development of a large variety of search techniques. In this paper we use the extreme-value approach as appears in Dekkers et al. [A moment estimator for the index of an extreme-value distribution, Ann. Statist. 17 (1989), pp. 1833–1855] and de Haan [Estimation of the minimum of a function using order statistics, J. Amer. Statist. Assoc. 76 (1981), pp. 467–469]. We impose some Lipschitz conditions on the functions being investigated and through repeated simulation-based samplings, we provide various practical interpretations of the parameters involved as well as point and interval estimates for x*.     

Nonparametric estimation of extreme conditional quantiles

The estimation of extreme conditional quantiles is an important issue in different scientific disciplines. Up to now, the extreme value literature focused mainly on estimation procedures based on independent and identically distributed samples. Our contribution is a two-step procedure for estimating extreme conditional quantiles. In a first step nonextreme conditional quantiles are estimated nonparametrically using a local version of [Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50.] regression quantile methodology. Next, these nonparametric quantile estimates are used as analogues of univariate order statistics in procedures for extreme quantile estimation. The performance of the method is evaluated for both heavy tailed distributions and distributions with a finite right endpoint using a small sample simulation study. A bootstrap procedure is developed to guide in the selection of an optimal local bandwidth. Finally the procedure is illustrated in two case studies.