Classical Extreme Value Theory Research Articles

Processes of rth largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an $\mathbb {R}^{\infty }$ -valued stochastic process in r which we denote as M(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M(r) and M(r) itself, after norming and centering. In continuous time, an analogous process Y(r) based on a two-dimensional Poisson process on $\mathbb {R}_{+}\times \mathbb {R}$ is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t > 0. This necessitates a different approach to the asymptotics in this case.

Моделирование полей экстремальных осадков на территорииЕвропейской части России

Present work is devoted to the study and development of space-time statistical structures ofextreme type modeling with the use of the max-stable processes. The theory of one-dimensionalextremal values and its extension to the two-dimensional case are considered and for that max-stable processes are introduced and then the main parametric families of max-stable processes(Schlather, Smith, Brown-Resnick, and Extremal-t) are presented. By modifying the maximumlikelihood method, namely using the paired likelihood function, parameter estimates wereobtained for each of the models whose efficiency was compared using the Takeuchi informationcriterion (TIC).Resulting models are coherent with classical extreme value theory and allow consistenttreatment of spatial dependence of rainfall. We illustrate the ideas through data, based ondaily cumulative rainfall totals recorded at 14 stations in central European part of Russia forperiod 1966-2016 years. We compare fits of different statistical models appropriate for spatialextremes and select the model that is the best for fitting our data. The method can be used inother situations to produce simulations needed for hydrological models, and in particular forthe generation of spatially heterogeneous extreme rainfall fields over catchments. It is shownthat the most successful model for the data we studied is the model from the extremal-t familywith the Whittle-Matern correlation function.

A latent trawl process model for extreme values

This paper presents a new model for characterizing temporal dependence in exceedances above a given threshold. Our model is based on a class of stationary, infinitely divisible stochastic processes known as trawl processes. For use with extreme values, our model is constructed by embedding a trawl process in a hierarchical framework. This ensures that the marginal distribution is a generalized Pareto, as expected from classical extreme value theory. We also consider a modified version of this model that works with a wider class of generalized Pareto distributions (GPDs) and has the advantage of separating marginal and temporal dependence properties. The model is illustrated via various applications to environmental time series; thus, we show that the model offers considerable flexibility in capturing the dependence structure of extreme value data.

Coupled Continuous Time Random Maxima

Continuous Time Random Maxima (CTRM) are a generalization of classical extreme value theory: Instead of observing random events at regular intervals in time, the waiting times between the events are also random variables which have arbitrary distributions. In case that the waiting times between the events have infinite mean, the limit process that appears differs from the limit process that appears in the classical case. With a continuous mapping approach, we derive a limit theorem for the case that the waiting times and the subsequent events are dependent as well as for the case that the waiting times depend on the preceding events (in this case we speak of an Overshooting Continuous Time Random Maxima, abbr. OCTRM). We get the distribution functions of the limit processes and a formula for the Laplace transform in time of the CTRM and the OCTRM limit. With this formula we have another way to calculate the distribution functions of the limit processes, namely by inversion of the Laplace transform. Moreover, we present governing equations which in our case are time fractional differential equations whose solutions are the distribution functions of our limit processes.

Conditional extreme value models: fallacies and pitfalls

Conditional extreme value models have been introduced by Heffernan and Resnick (Ann. Appl. Probab., 17, 537–571, 2007) to describe the asymptotic behavior of a random vector as one specific component becomes extreme. Obviously, this class of models is related to classical multivariate extreme value theory which describes the behavior of a random vector as its norm (and therefore at least one of its components) becomes extreme. However, it turns out that this relationship is rather subtle and sometimes contrary to intuition. We clarify the differences between the two approaches with the help of several illuminative (counter)examples. Furthermore, we discuss marginal standardization, which is a useful tool in classical multivariate extreme value theory but, as we point out, much less straightforward and sometimes even obscuring in conditional extreme value models. Finally, we indicate how, in some situations, a more comprehensive characterization of the asymptotic behavior can be obtained if the conditions of conditional extreme value models are relaxed so that the limit is no longer unique.

Future projections of Indian summer monsoon rainfall extremes over India with statistical downscaling and its consistency with observed characteristics

Indian summer monsoon rainfall extremes and their changing characteristics under global warming have remained a potential area of research and a topic of scientific debate over the last decade. This partially attributes to multiple definitions of extremes reported in the past studies and poor understanding of the changing processes associated with extremes. The later one results into poor simulation of extremes by coarse resolution General Circulation Models under increased greenhouse gas emission which further deteriorates due to inadequate representation of monsoon processes in the models. Here we use transfer function based statistical downscaling model with non-parametric kernel regression for the projection of extremes and find such conventional regional modeling fails to simulate rainfall extremes over India. In this conjuncture, we modify the downscaling algorithm by applying a robust regression to the gridded extreme rainfall events. We observe, inclusion of robust regression to the downscaling algorithm improves the historical simulation of rainfall extremes at a 0.25° spatial resolution, as evaluated based on classical extreme value theory methods, viz., block maxima and peak over threshold. The future projections of extremes during 2081–2100, obtained with the developed algorithm show no change to slight increase in the spatial mean of extremes with dominance of spatial heterogeneity. These changing characteristics in future are consistent with the observed recent changes in extremes over India. The proposed methodology will be useful for assessing the impacts of climate change on extremes; specifically while spatially mapping the risk to rainfall extremes over India.

Rare Events, Temporal Dependence, and the Extremal Index

Classical extreme-value theory for stationary sequences of random variables can up to a large extent be paraphrased as the study of exceedances over a high threshold. A special role within the description of the temporal dependence between such exceedances is played by the extremal index. Parts of this theory can be generalized not only to random variables on an arbitrary state space hitting certain failure sets but even to a triangular array of rare events on an abstract probability space.In the case of M4 processes, or maxima of multivariate moving maxima, the arguments take a simple and direct form.

Heavy tailed time series with extremal independence

We consider heavy tailed time series whose finite-dimensional distributions are extremally independent in the sense that extremely large values cannot be observed consecutively. This calls for methods beyond the classical multivariate extreme value theory which is convenient only for extremally dependent multivariate distributions. We use the Conditional Extreme Value approach to study the effect of an extreme value at time zero on the future of the time series. In formal terms, we study the limiting conditional distribution of future observations given an extreme value at time zero. To this purpose, we introduce conditional scaling functions and conditional scaling exponents. We compute these quantities for a variety of models, including Markov chains, exponential autoregressive models, stochastic volatility models with heavy tailed innovations or volatilities.

Statistics of Heteroscedastic Extremes

SummaryWe 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.

Extreme Value Laws for Superstatistics

We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fréchet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely χ2-superstatistics, inverse χ2 -superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that χ2-superstatistics generally leads an extreme value statistics described by a Fréchet distribution, whereas inverse χ2 -superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.

An improved estimator of the distortion risk measure for heavy-tailed claims

The main aim of this paper is to propose an alternative estimate of the distortion risk measure for heavy-tailed claims. Our approach is based on the result of Balkema and de Haan (1974) [3], and Pickands (1975) [22] for approximating the tail of the distribution by a generalized Pareto distribution. The asymptotic normality of the new estimator is established, and its performance illustrated by some results of simulation who shows the advantages of the new estimator over the estimator based on the classical extreme-value theory.

A sum characterization of hidden regular variation with likelihood inference via expectation-maximization

A fundamental deficiency of classical multivariate extreme value theory is the inability to distinguish between asymptotic independence and exact independence. In this work, we examine multivariate threshold modelling in the framework of regular variation on cones. Tail dependence is described by a limiting measure, which in some cases is degenerate on joint tail regions despite strong subasymptotic dependence in such regions. Hidden regular variation, a higher-order tail decay on these regions, offers a refinement of the classical theory. We develop a representation of random vectors possessing hidden regular variation as the sum of independent regular varying components. The representation is shown to be asymptotically valid via a multivariate tail equivalence result. We develop a likelihood-based estimation procedure from this representation via a Monte Carlo expectation-maximization algorithm which has been modified for tail estimation. The method is demonstrated on simulated data and applied to air pollution measurements.

Statistics of Heteroscedastic Extremes

We extend classical extreme value theory to non-identically distributed observations. When the distribution tails are proportional much of extreme value statistics remains valid. The proportionality function for the tails can be estimated nonparametrically along with the (common) extreme value index. Joint asymptotic normality of both estimators is shown; they are asymptotically independent. We develop tests for the proportionality function and for the validity of the model. We show through simulations the good performance of tests for tail homoscedasticity. The results are applied to stock market returns. A main tool is the weak convergence of a weighted sequential tail empirical process.

Rainfall extremes: Toward reconciliation after the battle of distributions.

[1] This study attempts to reconcile the conflicting results reported in the literature concerning the behavior of peak-over-threshold (POT) daily rainfall extremes and their distribution. By using two worldwide data sets, the impact of threshold selection and record length on the upper tail behavior of POT observations is investigated. The rainfall process is studied within the framework of generalized Pareto (GP) exceedances according to the classical extreme value theory (EVT), with particular attention paid to the study of the GP shape parameter, which controls the heaviness of the upper tail of the GP distribution. A twofold effect is recognized. First, as the threshold decreases, and nonextreme values are progressively incorporated in the POT samples, the variance of the GP shape parameter reduces and the mean converges to positive values denoting a tendency to heavy tail behavior. Simultaneously, the EVT asymptotic hypotheses are less and less realistic, and the GP asymptote tends to be replaced by the Weibull penultimate asymptote whose upper tail is exponential but apparently heavy. Second, for a fixed high threshold, the variance of the GP shape parameter reduces as the record length (number of years) increases, and the mean values tend to be positive, thus denoting again the prevalence of heavy tail behavior. In both cases, i.e., threshold selection and record length effect, the heaviness of the tail may be ascribed to mechanisms such as the blend of extreme and nonextreme values, and fluctuations of the parent distributions. It is shown how these results provide a link between previous studies and pave the way for more comprehensive analyses which merge empirical, theoretical, and operational points of view. This study also provides several ancillary results, such as a set of formulae to correct the bias of the GP shape parameter estimates due to short record lengths accounting for uncertainty, thus avoiding systematic underestimation of extremes which results from the analysis of short time series.Citation: Serinaldi, F., and C. G. Kilsby (2014), Rainfall extremes: Toward reconciliation after the battle of distributions, Water Resour. Res., 50, 336–352, doi:10.1002/2013WR014211.

Estimation of tail-related value-at-risk measures: range-based extreme value approach

This study proposes a new approach for estimating value-at-risk (VaR). This approach combines quasi-maximum-likelihood fitting of asymmetric conditional autoregressive range (ACARR) models to estimate the current volatility and classical extreme value theory (EVT) to estimate the tail of the innovation distribution of the ACARR model. The proposed approach reflects two well-known phenomena found in most financial time series: stochastic volatility and the fat-tailedness of conditional distributions. This approach presents two main advantages over the McNeil and Frey approach. First, the ACARR model in this approach is an asymmetric model that treats the upward and downward movements of the asset price asymmetrically, whereas the generalized autoregressive conditional heteroskedasticity model in the McNeil and Frey approach is a symmetric model that ignores the asymmetric structure of the asset price. Second, the proposed method uses classical EVT to estimate the tail of the distribution of the residuals to avoid the threshold issue in the modern EVT model. Since the McNeil and Frey approach uses modern EVT, it may estimate the tail of the innovation distribution poorly. Back testing of historical time series data shows that our approach gives better VaR estimates than the McNeil and Frey approach.

Importance of Generalized Logistic Distribution in Extreme Value Modeling

We consider a problem from stock market modeling, precisely, choice of adequate distribution of modeling extremal behavior of stock market data. Generalized extreme value (GEV) distribution and generalized Pareto (GP) distribution are the classical distributions for this problem. However, from 2004, [1] and many other researchers have been empirically showing that generalized logistic (GL) distribution is a better model than GEV and GP distributions in modeling extreme movement of stock market data. In this paper, we show that these results are not accidental. We prove the theoretical importance of GL distribution in extreme value modeling. For proving this, we introduce a general multivariate limit theorem and deduce some important multivariate theorems in probability as special cases. By using the theorem, we derive a limit theorem in extreme value theory, where GL distribution plays central role instead of GEV distribution. The proof of this result is parallel to the proof of classical extremal types theorem, in the sense that, it possess important characteristic in classical extreme value theory, for e.g. distributional property, stability, convergence and multivariate extension etc.

Statistical analysis on extreme wave height

The classical extreme value theory based on generalized extreme value (GEV) distribution and generalized Pareto distribution (GPD) is applied to the wave height estimate based on wave hindcast data covering a period of 31 years for a location in the eastern Arabian Sea. Practical concern such as the threshold selection and model validation is discussed. Estimates of wave height having different return periods are compared with estimates obtained from different distributions. On comparing the distributions fitted to the GEV with annual maximum approach and GPD with peaks over threshold approach have indicated that both GEV and GPD models gave similar or comparable wave height for the study area since there is no multiple storm event in a year. Influence of seasonality on wave height having different return period is also studied.

On the limit distribution of lower extreme generalized order statistics

In a wide subclass of generalized order statistics (gOs), which contains most of the known and important models of ordered random variables, weak convergence of lower extremes are developed. A recent result of extreme value theory of m −gOs (as well as the classical extreme value theory of ordinary order statistics) yields three types of limit distributions that are possible in case of linear normalization. In this paper a similar classification of limit distributions holds for extreme gOs, where the parameters γj,j = 1, . . . , n, are assumed to be pairwise different. Two illustrative examples are given to demonstrate the practical importance for some of the obtained results.

Extreme value analysis of actuarial risks: estimation and model validation

We give an overview of several aspects arising in the statistical analysis of extreme risks with actuarial applications in view. In particular it is demonstrated that empirical process theory is a very powerful tool, both for the asymptotic analysis of extreme value estimators and to devise tools for the validation of the underlying model assumptions. While the focus of the paper is on univariate tail risk analysis, the basic ideas of the analysis of the extremal dependence between different risks are also outlined. Here we emphasize some of the limitations of classical multivariate extreme value theory and sketch how a different model proposed by Ledford and Tawn can help to avoid pitfalls. Finally, these theoretical results are used to analyze a data set of large claim sizes from health insurance.

Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. [2006] have found analytical results.