Extreme Value Index Research Articles

Extremal quantile autoregression for heavy-tailed time series

The conventional quantile estimator of the extreme conditional quantiles under quantile autoregression models is relatively unstable due to data sparsity in the tails. Extreme value theory provides a mathematical foundation of extrapolation to estimate extreme quantiles. However, the asymptotic distributions of existing estimators are often complicated making it inconvenient to apply for statistical inference. We develop a new adaptive estimation procedure based on a generalized Hill estimator of the extreme value index to estimate extreme conditional quantiles in autoregression models. We establish the asymptotic normality of the proposed estimators under some regularity conditions by applying the martingale central limit theorem. The asymptotic variances have closed expressions and can be directly estimated. Based on these results, we propose a procedure to construct confidence intervals for the extreme conditional quantiles. We also provide a diagnostic tool to check the heavy-tailedness of the distribution. Simulation studies and a real data analysis are carried out to demonstrate the advantages of the proposed methods over existing approaches.

The Use of Generalized Means in the Estimation of the Weibull Tail Coefficient

Due to the specificity of the Weibull tail coefficient, most of the estimators available in the literature are based on the log excesses and are consequently quite similar to the estimators used for the estimation of a positive extreme value index. The interesting performance of estimators based on generalized means leads us to base the estimation of the Weibull tail coefficient on the power mean-of-order-p. Consistency and asymptotic normality of the estimators under study are put forward. Their performance for finite samples is illustrated through a Monte Carlo simulation. It is always possible to find a negative value ofp(contrarily to what happens with the mean-of-order-pestimator for the extreme value index), such that, for adequate values of the threshold, there is a reduction in both bias and root mean square error.

Non-regular Frameworks and the Mean-of-Order p Extreme Value Index Estimation

Most of the estimators of parameters of rare and large events, among which we distinguish the extreme value index (EVI) for maxima, one of the primary parameters in statistical extreme value theory, are averages of statistics, based on the k upper observations. They can thus be regarded as the logarithm of the geometric mean, i.e. the logarithm of the power mean of order \(p=0\) of a certain set of statistics. Only for heavy tails, i.e. a positive EVI, quite common in many areas of application, and trying to improve the performance of the classical Hill EVI-estimators, instead of the aforementioned geometric mean, we can more generally consider the power mean of order-p (MO\(_p\)) and build associated MO\(_p\) EVI-estimators. The normal asymptotic behaviour of MO\(_p\) EVI-estimators has already been obtained for \(p<1/(2\xi )\), with consistency achieved for \(p<1/\xi \), where \(\xi \) denotes the EVI. We shall now consider the non-regular case, \(p\ge 1/(2\xi )\), a situation in which either normal or non-normal sum-stable laws can be obtained, together with the possibility of an ‘almost degenerate’ EVI-estimation.

Adapting the Hill estimator to distributed inference: dealing with the bias

The distributed Hill estimator is a divide-and-conquer algorithm for estimating the extreme value index when data are stored in multiple machines. In applications, estimates based on the distributed Hill estimator can be sensitive to the choice of the number of the exceedance ratios used in each machine. Even when choosing the number at a low level, a high asymptotic bias may arise. We overcome this potential drawback by designing a bias correction procedure for the distributed Hill estimator, which adheres to the setup of distributed inference. The asymptotically unbiased distributed estimator we obtained, on the one hand, is applicable to distributed stored data, on the other hand, inherits all known advantages of bias correction methods in extreme value statistics.

The Stochastic Approximation Method for Recursive Kernel Estimation of the Conditional Extreme Value Index

In this research paper, we apply the stochastic approximation method to define a class of recursive kernel estimator of the conditional extreme value index. We investigate the properties of the proposed recursive estimator and compare them to those pertaining to Hill’s non-recursive kernel estimator. We show that using some optimal parameters, the proposed recursive estimator defined by the stochastic approximation algorithm proves to be very competitive to Hill’s non-recursive kernel estimator. Finally, the theoretical results are confirmed through simulation experiments and illustrated using a real dataset about Malaria in Senegalese children.

Box-Cox Transformations and Bias Reduction in Extreme Value Theory

The Box-Cox transformations are used to make the data more suitable for statistical analysis. We know from the literature that this transformation of the data can increase the rate of convergence of the tail of the distribution to the generalized extreme value distribution, and as a byproduct, the bias of the estimation procedure is reduced. The reduction of bias of the Hill estimator has been widely addressed in the literature of extreme value theory. Several techniques have been used to achieve such reduction of bias, either by removing the main component of the bias of the Hill estimator of the extreme value index (EVI) or by constructing new estimators based on generalized means or norms that generalize the Hill estimator. We are going to study the Box-Cox Hill estimator introduced by Teugels and Vanroelen, in 2004, proving the consistency and asymptotic normality of the estimator and addressing the choice and estimation of the power and shift parameters of the Box-Cox transformation for the EVI estimation. The performance of the estimators under study will be illustrated for finite samples through small-scale Monte Carlo simulation studies.

Detection of increase in air temperature in Barrow, AK, USA, through the use of extreme value indices and its impact on the permafrost active layer thickness

Detection of increase in air temperature in Barrow, AK, USA, through the use of extreme value indices and its impact on the permafrost active layer thickness

A Different Way of Choosing a Threshold in a Bivariate Extreme Value Study

The choice of optimum threshold in Extreme Value Theory, peaks over threshold, has been a topic of discussionfor decades. A threshold must be chosen high enough to control the bias of the extreme value index. On the other hand, if a threshold is chosen too high the variance becomes a problem. This is a very difficult trade-off and has been studied over the years from various viewpoints. More often these studies aim at methods for choosing the threshold in univariate settings. Not as many literature are available for choosing the threshold in a multivariate setting. In this paper we consider an approach for choosing the threshold when working with bivariate extreme values above a threshold. This approach makes use of Bayesian methodology. It adds value to the existing literature since it is also possible to use this approach without visual inspection.

Testing for changes in the tail behavior of Brown–Resnick Pareto processes

We consider the class of r-Pareto processes defined on [0,1] whose max-stable counterparts are Brown–Resnick processes. The aim of this paper is to propose a test whether the extreme value index function of a r-Pareto process of this class remains constant over [0,1]. We assume that we observe several independent r-Pareto processes over equispaced grids of [0,1] but with possibly heterogeneous grid resolutions. We build a test based on the normalized approximate total variations of the paths of these processes. We provide its properties under infill asymptotics and assess its finite sample performance through simulation experiments. We discuss how to proceed for random processes that belong to the domain of attraction of Brown–Resnick r-Pareto processes and illustrate the approach with an application to wind speed data from Germany.

Extreme quantile regression for tail single-index varying-coefficient models

This paper studies the estimation of extreme conditional quantile for single-index varying-coefficient models (SIVCM). Our contribution is a three-step extreme quantile regression procedure for SIVCM. In the first step, we estimate the intermediate conditional quantile in a quantile regression framework. In the second step, we estimate the extreme value index (EVI) based on the estimated intermediate quantile. In the third step, we extrapolate the intermediate conditional quantile to the extreme tails by adapting the extreme value theory (EVT). We obtain the asymptotation of tail single-index varying-coefficient estimators and demonstrate through simulation study and real data example that the proposed methods enjoy higher accuracy than the conventional quantile regression estimates.

A class of location invariant estimators for heavy tailed distributions

In this paper, a new class of location-invariant semi-parametric estimators of a positive extreme value index is proposed. Its asymptotic distributional representation and asymptotic normality are derived, and the optimal choice of the sample fraction by mean squared error is also discussed for some special cases. Finally comparison studies are provided for some familiar models by Monte Carlo simulations.

A class of weighted Hill estimators

In Statistics of Extremes, the estimation of the extreme value index is an essential requirement for further tail inference. In this work, we deal with the estimation of a strictly positive extreme value index from a model with a Pareto-type right tail. Under this framework, we propose a new class of weighted Hill estimators, parameterized with a tuning parameter a. We derive their non-degenerate asymptotic behavior and analyze the influence of the tuning parameter in such result. Their finite sample performance is analyzed through a Monte Carlo simulation study. A comparison with other important extreme value index estimators from the literature is also provided.

Threshold selection in univariate extreme value analysis

Threshold selection plays a key role in various aspects of statistical inference of rare events. In this work, two new threshold selection methods are introduced. The first approach measures the fit of the exponential approximation above a threshold and achieves good performance in small samples. The second method smoothly estimates the asymptotic mean squared error of the Hill estimator and performs consistently well over a wide range of processes. Both methods are analyzed theoretically, compared to existing procedures in an extensive simulation study and applied to a dataset of financial losses, where the underlying extreme value index is assumed to vary over time.

Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications

In this paper, we introduce a new robust estimator for the extreme value index of Pareto-type distributions under randomly right-truncated data and establish its consistency and asymptotic normality. Our considerations are based on the Lynden-Bell integral and a useful huberized M-functional and M-estimators of the tail index. A simulation study is carried out to evaluate the robustness and the nite sample behavior of the proposed estimator. Extreme quantiles estimation is also derived and applied to real data-set of lifetimes of automobile brake pads.

Distributed inference for the extreme value index

Summary In this paper we investigate a divide-and-conquer algorithm for estimating the extreme value index when data are stored in multiple machines. The oracle property of such an algorithm based on extreme value methods is not guaranteed by the general theory of distributed inference. We propose a distributed Hill estimator and establish its asymptotic theories. We consider various cases where the number of observations involved in each machine can be either homogeneous or heterogeneous, and either fixed or varying according to the total sample size. In each case we provide a sufficient, sometimes also necessary, condition under which the oracle property holds.

Functional kernel estimation of the conditional extreme value index under random right censoring

Estimation of the extreme-value index of a heavy-tailed distribution is investigated when some functional random covariate (i.e. valued in some infinite dimensional space) information is available and the scalar response variable is right-censored. A weighted kernel version of Hill’s estimator of the extreme-value index is proposed and its asymptotic normality is established under mild assumptions.A simulation study is conducted to assess the finite-sample behavior of the proposed estimator. An application to ambulatory blood pressure trajectories and clinical outcome in stroke patients is also provided.

Robust Estimator of Conditional Tail Expectation of Pareto-Type Distribution

In this paper, we use the extreme value index estimator, called the t-Hill, to derive a robust estimator of conditional tail expectation (CTE) in the case of heavy-tailed losses. The CTE is rapidly turning into the favored measure for statutory assessment of the balance sheet at whatever point true stochastic techniques are utilized to fix the provisions for risks. Under the extreme value methodology, we prove the asymptotic normality of the robust nonparametric conditional tail expectation estimator when the loss variable follows any distribution with infinite second moment. In addition, the numerical performance of our new estimator is studied and compared to a well-established estimator, with favorable results.

Trimmed extreme value estimators for censored heavy-tailed data

We consider estimation of the extreme value index and extreme quantiles for heavy–tailed data that are right-censored. We study a general procedure of removing low importance observations in tail estimators. This trimming procedure is applied to the state-of-the-art estimators for randomly right-censored tail estimators. Through an averaging procedure over the amount of trimming we derive new kernel type estimators. Extensive simulation suggests that one of the new considered kernels leads to a highly competitive estimator against virtually any other available alternative in this framework. Moreover we propose an adaptive selection method for the amount of top data used in estimation based on the trimming procedure minimizing the asymptotic mean squared error. We also provide an illustration of this approach to simulated as well as to real-world MTPL insurance data.

EXTREME VALUE STATISTICS IN SEMI-SUPERVISED MODELS

We consider extreme value analysis in a semi-supervised setting, where we observe, next to the n data on the target variable, n +m data on one or more covariates. This is called the semi-supervised model with n labeled and m unlabeled data. By exploiting the tail dependence between the target variable and the covariates, we derive an estimator for the extreme value index of the target variable in this setting and establish its asymptotic behavior. Our estimator substantially improves the univariate estimator, based on only the n target variable data, in terms of asymptotic variance whereas the asymptotic bias remains unchanged. We present a simulation study in which the asymptotic results are confirmed and also an extreme quantile estimator is derived and its improved performance is shown. Finally the estimation method is applied to rainfall data in France.

Functional Kernel Estimation of the Conditional Extreme Quantile under Random Right Censoring

The study of estimation of conditional extreme quantile in incomplete data frameworks is of growing interest. Specially, the estimation of the extreme value index in a censorship framework has been the purpose of many investigations when finite dimension covariate information has been considered. In this paper, the estimation of the conditional extreme quantile of a heavy-tailed distribution is discussed when some functional random covariate (i.e. valued in some infinite-dimensional space) information is available and the scalar response variable is right-censored. A Weissman-type estimator of conditional extreme quantiles is proposed and its asymptotic normality is established under mild assumptions. A simulation study is conducted to assess the finite-sample behavior of the proposed estimator and a comparison with two simple estimations strategies is provided.