The Hill estimators under power normalization

By using the link between the affine and the power norming, eight Hill estimators under power normalization for the tail index (the non-zero extreme value index) are suggested. Moreover, more compact and adaptive four Hill estimators under power normalization are derived based on the generalized Pareto distributions under power normalization. Two classes of harmonic t-Hill estimators under power normalization are also suggested. A comprehensive simulation study using the R-package shows that all the suggested estimators under power normalization work well, but in all cases the Hill estimators under power normalization based on Pareto distributions under power normalization are better. The two models under linear and power normalization for extreme value analysis are applied with comparison on a real data set of two pollutants, Sulphur Dioxide and Particulate Matter.     

A robust and efficient adaptive reweighted estimator of multivariate location and scatter

This article proposes a reweighted estimator of multivariate location and scatter, with weights adaptively computed from the data. Its breakdown point and asymptotic behavior under elliptical distributions are established. This adaptive estimator is able to attain simultaneously the maximum possible breakdown point for affine equivariant estimators and full asymptotic efficiency at the multivariate normal distribution. For the special case of hard-rejection weights and the MCD as initial estimator, it is shown to be more efficient than its non-adaptive counterpart for a broad range of heavy-tailed elliptical distributions. A Monte Carlo study shows that the adaptive estimator is as robust as its non-adaptive relative for several types of bias-inducing contaminations, while it is remarkably more efficient under normality for sample sizes as small as 200.     

一类新的位置不变矩估计及其渐近正态性

The moment estimator has been widely used in extreme value theory in order to estimate the extreme value index, however it is not location invariant. In this paper, based on the moment-type estimator, we propose a new location invariant moment-type estimator,and discuss its asymptotic normality under the second order regular variation. Finally, a simulation is presented to compare this new estimator with another location invariant momenttype estimator γ_n~M(k_0, k) proposed by Ling, which indicates that the new estimator has good performances.     

Likelihood Based Confidence Intervals for the Tail Index

For the estimation of the tail index of a heavy tailed distribution, one of the well-known estimators is the Hill estimator (Hill, 1975). One obvious way to construct a confidence interval for the tail index is via the normal approximation of the Hill estimator. In this paper we apply both the empirical likelihood method and the parametric likelihood method to obtaining confidence intervals for the tail index. Our limited simulation study indicates that the normal approximation method is worse than the other two methods in terms of coverage probability, and the empirical likelihood method and the parametric likelihood method are comparable.     

A Deterministic Algorithm for Robust Location and Scatter

Most algorithms for highly robust estimators of multivariate location and scatter start by drawing a large number of random subsets. For instance, the FASTMCD algorithm of Rousseeuw and Van Driessen starts in this way, and then takes so-called concentration steps to obtain a more accurate approximation to the MCD. The FASTMCD algorithm is affine equivariant but not permutation invariant. In this article, we present a deterministic algorithm, denoted as DetMCD, which does not use random subsets and is even faster. It computes a small number of deterministic initial estimators, followed by concentration steps. DetMCD is permutation invariant and very close to affine equivariant. We compare it to FASTMCD and to the OGK estimator of Maronna and Zamar. We also illustrate it on real and simulated datasets, with applications involving principal component analysis, classification, and time series analysis. Supplemental material (Matlab code of the DetMCD algorithm and the datasets) is available online.     

Estimation of the extreme value index and extreme quantiles under random censoring

In this paper, we consider the estimation of the extreme value index and extreme quantiles in the presence of random right censoring. The generalization of the peaks over threshold method is discussed and an adaptation of the moment estimator is proposed. The corresponding extreme quantile estimators are also introduced. We make a start with the analysis of the asymptotic properties of the moment estimator and the corresponding extreme quantile estimator. The finite sample behaviour is illustrated with a small simulation study and through practical examples from survival data analysis.      

A Location Invariant Hill-Type Estimator

The Hill's estimator (Hill, 1975) has been largely used in extreme value theory in order to estimate the tail index associated to a distribution function with a positive index. One of the criticisms to its use is the possible associated bias, dependent on the top portion of the original sample used, and also the fact that it is not location invariant. Here, a new Hill-type estimator is studied, which is location invariant. This new estimator is based on the original Hill's estimator, but is made location invariant by a random shift. Its asymptotic distributional behavior is derived, in a semiparametric setup. A comparative simulation study is also presented for several models, following an appropriate adaptive procedure.     

Asymptotic normality of location invariant heavy tail index estimator

Motivated by Fraga Alves (Extremes 4:199–217, 2001)’s work, a new class of location invariant Hill-type estimators for the tail index of a heavy tailed distribution is proposed in the paper. Its asymptotic behavior is derived, and the optimal choice of the sample fraction is discussed by mean squared error. Asymptotic comparisons and simulation studies are presented to show that the new estimator performs well compared to the known ones.     

A new test for multivariate normality

We propose a new class of rotation invariant and consistent goodness-of-fit tests for multivariate distributions based on Euclidean distance between sample elements. The proposed test applies to any multivariate distribution with finite second moments. In this article we apply the new method for testing multivariate normality when parameters are estimated. The resulting test is affine invariant and consistent against all fixed alternatives. A comparative Monte Carlo study suggests that our test is a powerful competitor to existing tests, and is very sensitive against heavy tailed alternatives.     

Weak properties and robustness of t-Hill estimators

We describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. Here, we study both forms of the estimator, i.e. t-Hill and t-lgHill. For both estimators we prove weak consistency in moving average settings as well as the asymptotic normality of t-lgHill estimator in iid setting. In cases of contamination with heavier tails than the tail of original sample, t-Hill outperforms several robust tail estimators, especially in small samples. A simulation study emphasizes the fact that the level of contamination is playing a crucial role. The larger the contamination, the better are the t-score moment estimates. The reason for this is the bounded t-score of heavy-tailed distributions (and, consequently, bounded influence functions of the estimators). We illustrate the developed methodology on a small sample data set of stake measurements from Guanaco glacier in Chile.     

The finite sample breakdown point of PCS

The Projection Congruent Subset (PCS) is a new method for finding multivariate outliers. PCS returns an outlyingness index which can be used to construct affine equivariant estimates of multivariate location and scatter. In this note, we derive the finite sample breakdown point of these estimators.     

The influence function of the Stahel–Donoho estimator of multivariate location and scatter

This article derives the influence function of the Stahel–Donoho estimator of multivariate location and scatter for elliptical distributions. Local robustness and asymptotic relative efficiency are studied. The expressions obtained for the influence functions coincide with those of one-step reweighted estimators.     

Alternatives to a Semi-Parametric Estimator of Parameters of Rare Events—The Jackknife Methodology*

In this paper, and in a context of regularly varying tails, we propose different alternatives to a well-known estimator of the tail index—the Hill estimator (Hill, 1975). These alternatives have essentially in mind a reduction in bias, preferably without increasing Mean Square Error, by the use of suitable Generalized Jackknife methodologies (Gray and Schucany, 1972). The first estimate obtained through this methodolgy is the one introduced by Peng (1998), under a different context. Other Generalized Jackknife estimators are linear combinations of Hill estimators at different levels. This methodology of affine combinations of Hill estimators at different levels may be easily generalized to other semi-parametric estimators of the tail index, like Pickands' estimator (Pickands, 1975) or the Moment's estimator (Dekkers et al., 1989), and consequently to a general real tail index, seeming to be a promising field of research.     

Bias correction in extreme value statistics with index around zero

Applying extreme value statistics in meteorology and environmental science requires accurate estimators on extreme value indices that can be around zero. Without having prior knowledge on the sign of the extreme value indices, the probability weighted moment (PWM) estimator is a favorable candidate. As most other estimators on the extreme value index, the PWM estimator bears an asymptotic bias. In this paper, we develop a bias correction procedure for the PWM estimator. Moreover, we provide bias-corrected PWM estimators for high quantiles and, when the extreme value index is negative, the endpoint of a distribution. The choice of k, the number of high order statistics used for estimation, is crucial in applications. The asymptotically unbiased PWM estimators allows the choice of higher level k, which results in a lower asymptotic variance. Moreover, since the bias-corrected PWM estimators can be applied for a wider range of k compared to the original PWM estimator, one gets more flexibility in choosing k for finite sample applications. All advantages become apparent in simulations and an environmental application on estimating “once per 10,000 years” still water level at Hoek van Holland, The Netherlands.     

Estimation of limiting conditional distributions for the heavy tailed long memory stochastic volatility process

We consider Stochastic Volatility processes with heavy tails and possible long memory in volatility. We study the limiting conditional distribution of future events given that some present or past event was extreme (i.e. above a level which tends to infinity). Even though extremes of stochastic volatility processes are asymptotically independent (in the sense of extreme value theory), these limiting conditional distributions differ from the i.i.d. case. We introduce estimators of these limiting conditional distributions and study their asymptotic properties. If volatility has long memory, then the rate of convergence and the limiting distribution of the centered estimators can depend on the long memory parameter (Hurst index).     

Estimation of a tail index based on minimum density power divergence

In this paper, we consider the minimum density power divergence estimator for the tail index of heavy tailed distributions in strong mixing processes. It is shown that the estimator is consistent and asymptotically normal under regularity conditions. The simulation results demonstrate that the estimator is robust in the presence of outliers.     

Robust weighted orthogonal regression in the errors-in-variables model

This paper focuses on robust estimation in the structural errors-in-variables (EV) model. A new class of robust estimators, called weighted orthogonal regression estimators, is introduced. Robust estimators of the parameters of the EV model are simply derived from robust estimators of multivariate location and scatter such as the M-estimators, the S-estimators and the MCD estimator. The influence functions of the proposed estimators are calculated and shown to be bounded. Moreover, we derive the asymptotic distributions of the estimators and illustrate the results on simulated examples and on a real-data set.     

Reduced‐bias kernel estimators of a positive extreme value index

In this paper, we deal with the semi‐parametric estimation of the extreme value index, an important parameter in extreme value analysis. It is well known that many classic estimators, such as the Hill estimator, reveal a strong bias. This problem motivated the study of two classes of kernel estimators. Those classes generalize the classical Hill estimator and have a tuning parameter that enables us to modify the asymptotic mean squared error and eventually to improve their efficiency. Since the improvement in efficiency is not very expressive, we also study new reduced bias estimators based on the two classes of kernel statistics. Under suitable conditions, we prove their asymptotic normality. Moreover, an asymptotic comparison, at optimal levels, shows that the new classes of reduced bias estimators are more efficient than other reduced bias estimator from the literature. An illustration of the finite sample behaviour of the kernel reduced‐bias estimators is also provided through the analysis of a data set in the field of insurance.     

Symmetrised M-estimators of multivariate scatter

In this paper we introduce a family of symmetrised M-estimators of multivariate scatter. These are defined to be M-estimators only computed on pairwise differences of the observed multivariate data. Symmetrised Huber's M-estimator and Dümbgen's estimator serve as our examples. The influence functions of the symmetrised M-functionals are derived and the limiting distributions of the estimators are discussed in the multivariate elliptical case to consider the robustness and efficiency properties of estimators. The symmetrised M-estimators have the important independence property; they can therefore be used to find the independent components in the independent component analysis (ICA).     

一类位置不变的Pickands型估计量

当极值指标大于0时, 本文提出了一种位置不变的Pickands型估计量,证明了该估计量的强弱相合性, 给出了其渐近展式和强收敛速度,并对$k_2$的最优选择进行了讨论,最后利用自适应性方法对该估计量和其它Pickands型估计量进行随机模拟分析,比较该估计量的优越性.