Pickands Estimator Research Articles

Tail inference using extreme U-statistics

Extreme U-statistics arise when the kernel of a U-statistic has a high degree but depends only on its arguments through a small number of top order statistics. As the kernel degree of the U-statistic grows to infinity with the sample size, estimators built out of such statistics form an intermediate family in between those constructed in the block maxima and peaks-over-threshold frameworks in extreme value analysis. The asymptotic normality of extreme U-statistics based on location-scale invariant kernels is established. Although the asymptotic variance coincides with the one of the Hájek projection, the proof goes beyond considering the first term in Hoeffding’s variance decomposition. We propose a kernel depending on the three highest order statistics leading to a location-scale invariant estimator of the extreme value index resembling the Pickands estimator. This extreme Pickands U-estimator is asymptotically normal and its finite-sample performance is competitive with that of the pseudo-maximum likelihood estimator.

Bivariate nonparametric estimation of the Pickands dependence function using Bernstein copula with kernel regression approach

In this study, a new nonparametric approach using Bernstein copula approximation is proposed to estimate Pickands dependence function. New data points obtained with Bernstein copula approximation serve to estimate the unknown Pickands dependence function. Kernel regression method is then used to derive an intrinsic estimator satisfying the convexity. Some extreme-value copula models are used to measure the performance of the estimator by a comprehensive simulation study. Also, a real-data example is illustrated. The proposed Pickands estimator provides a flexible way to have a better fit and has a better performance than the conventional estimators.

Estimation of high conditional quantiles using the Hill estimator of the tail index

To implement the extremal quantile regression, one needs to have an accurate estimate of the tail index that is involved in the limit distributions of extremal regression quantiles. However, the existing quantile estimation methods are often unstable owing to data sparsity in the tails. In this paper, we propose the Hill estimator for the tail index based on regression quantiles, and construct a new estimator for high conditional quantiles through an extrapolation of the intermediate regression quantiles. In both theory and simulation, we demonstrate that the proposed estimators are more efficient than those based on the refined Pickands estimator of the tail index. The applicability of the new method is also illustrated on the Occidental Petroleum daily stock return data.

Optimally robust estimators in generalized Pareto models

In this paper, we study the robustness properties of several procedures for the joint estimation of shape and scale in a generalized Pareto model. The estimators that we primarily focus upon, most bias robust estimator (MBRE) and optimal MSE-robust estimator (OMSE), are one-step estimators distinguished as optimally robust in the shrinking neighbourhood setting; that is, they minimize the maximal bias, respectively, on such a specific neighbourhood, the maximal mean squared error (MSE). For their initialization, we propose a particular location–dispersion estimator, MedkMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally robust estimators are compared to the maximum-likelihood, skipped maximum-likelihood, Cramér–von-Mises minimum distance, method-of-medians, and Pickands estimators. To quantify their deviation from robust optimality, for each of these suboptimal estimators, we determine the finite-sample breakdown point and the influence function, as well as the statistical accuracy measured by asymptotic bias, variance, and MSE – all evaluated uniformly on shrinking neighbourhoods. These asymptotic findings are complemented by an extensive simulation study to assess the finite-sample behaviour of the considered procedures. The applicability of the procedures and their stability against outliers are illustrated for the Danish fire insurance data set from the package evir.

Asymptotic properties of generalized DPR statistic

We investigate asymptotic properties of the so-called generalized Davydov, Paulauskas, and Rackauskas statistic that was introduced in [V. Paulauskas and M. Vaiciulis, Several modifications of DPR estimator of the tail index, Lith. Math. J., 51(1):36–50, 2011]. We obtain an asymptotics of the bias of this statistic and show that this statistic is asymptotically normal. As an application, we provide a new class of semiparametric heavy-tailed estimators and compare these estimators with other estimators, including the popular Hill and Pickands estimators.

A two-step estimator of the extreme value index

In this paper, we build a two-step estimator $\hat{\gamma}_{\rm STEP}$ , which satisfies $\sqrt{k}(\hat{\gamma}_{\rm STEP}-\hat{\gamma}_{ML})\stackrel{P}{\rightarrow} 0$ , where $\hat{\gamma}_{ML}$ is the well-known maximum likelihood estimator of the extreme value index. Since the two-step estimator $\hat{\gamma}_{\rm STEP}$ can be calculated easily as a function of the observations, it is much simpler to use in practice. By properly choosing the first step estimator, such as the Pickands estimator, we can even get a shift and scale invariant estimator with the above property.

Nonparametric estimation of the dependence function for a multivariate extreme value distribution

Understanding and modeling dependence structures for multivariate extreme values are of interest in a number of application areas. One of the well-known approaches is to investigate the Pickands dependence function. In the bivariate setting, there exist several estimators for estimating the Pickands dependence function which assume known marginal distributions [J. Pickands, Multivariate extreme value distributions, Bull. Internat. Statist. Inst., 49 (1981) 859–878; P. Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 (1991) 429–439; P. Hall, N. Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (2000) 835–844; P. Capéraà, A.-L. Fougères, C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (1997) 567–577]. In this paper, we generalize the bivariate results to p-variate multivariate extreme value distributions with p ⩾ 2 . We demonstrate that the proposed estimators are consistent and asymptotically normal as well as have excellent small sample behavior.

Non-Parametric Inference for Bivariate Extreme-Value Copulas

Extreme-value copulas arise as the possible limits of copulas of component-wise maxima of independent, identically distributed samples. The use of bivariate extreme-value copulas is greatly facilitated by their representation in terms of Pickands dependence functions. The two main families of estimators of this dependence function are (variants of) the Pickands estimator and the Caperaa-Fougeres-Genest estimator. In this paper, a unified treatment is given of these two families of estimators, and within these classes those estimators with the minimal asymptotic variance are determined. Main result is the explicit construction of an adaptive, minimum-variance estimator within a class of estimators that encompasses the Caperaa-Fougeres-Genest estimator.

Generalized Pickands estimators for the extreme value index

The Pickands estimator for the extreme value index is generalized in a way that includes all of its previously known variants. A detailed study of the asymptotic behavior of the estimators in the family serves to determine its optimally performing members. These are given by simple, explicit formulas, have the same asymptotic variance as the maximum likelihood estimator in the generalized Pareto model, and are robust to departures from the limiting generalized Pareto model in case the convergence of the excess distribution to its limit is slow. A simulation study involving a wide range of distributions shows the new estimators to compare favorably with the maximum likelihood estimator.

A class of Pickands-type estimators for the extreme value index

In this paper we introduce a new class of Pickands-type estimators for the extreme value index. Consistency and asymptotic normality of the estimators are established under suitable regularity conditions. The new estimator uses the same fraction of upper-order statistics as the well-known Pickands estimator, but its asymptotic performance is better by at least 10% if it is chosen in some optimal ways. A simulation study also demonstrates its good finite-sample performance.

Optimal rates of convergence for estimates of the extreme value index

Hall and Welsh established the best attainable rate of convergence for estimates of a positive extreme value index $\gamma$ under a certain second order condition implying that the distribution function of the maximum of n random variables converges at an algebraic rate to the pertaining extreme value distribution. As a first generalization, we obtain a surprisingly sharp bound on the estimation error if $\gamma$ is still assumed to be positive, but the rate of convergence of the maximum may be nonalgebraic. This result allows a more accurate evaluation of the asymptotic performance of an estimator for $\gamma$ than the Hall and Welsh theorem. For example, it is proved that the Hill and the Pickands estimators achieve the optimal rate, but only the Hill estimator attains the sharp bound. Finally, an analogous result is derived for a general, not necessarily positive, extreme value index. In this situation it turns out that location invariant estimators show the best performance.

Refined pickands estimators wtth bias correction

Consider an univariate distribution function F that belongs to the weak domain of attraction of an extreme value distribution. In Drees (1994) mixtures of Pickands estimators of the extreme value index β were considered that are based on the kn largest order statistics. It was shown that under certain second order conditions on F the estimator is asymptotically biased if kn grows too fast. Here we introduce a quite general bias correcting procedure that allows to utilize more largest order statistics. Moreover, it is proven that the estimator yields sensible results even if some of the model assumptions are not satisfied. A simulation study demonstrates the robustness against an unsuitable choice of kn.

Comparison of estimation methods in extreme value theory

In this study we compare three estimators of the extreme value index: Pickands estimator, the moment estimator and a maximum likelihood estimator. The estimators are explored both theoretically and by Monte Carlo simulation. We obtain two estimators for large quantiles using Pickands and the maximum likelihood estimators. The latter and one based on the moment estimator are then compared through simulation.

Refined Pickands estimators of the extreme value index

Consider a distribution function that belongs to the weak domain of attraction of an extreme value distribution. The extreme value index $\beta$ will be estimated by mixtures of Pickands estimators, where the weights are generated by a probability measure which satisfies a certain integrability condition. We prove a functional limit theorem for a process of Pickands estimators and asymptotic normality of the refined Pickands estimator. For negative $\beta$ the new estimator is asymptotically superior to previously defined estimators. A simulation study also demonstrates the good small-sample performance. In particular, the estimator proves to be robust against an inappropriate choice of the number of upper order statistics used for estimation.

Efficiency of convex combinations of pickands estimator of the extreme value index

Consider a distribution function F in the domain of attraction of an extreme value distribution G β with unknown extreme value index . An appealing nonparametric estimate of β is the Pickands estimator , which is based on the 4m largest observations in a sample of size n generated independently according to F. If F satisfies a von Mises condition with rapidly decreasing remainder term, we can establish asymptotic normality of convex combinations of Pickands estimate. With the asymptotically optimal p = p opt∊[0,1] minimizing the limiting variance of Pickands estimator is then clearly outperformed by the convex combination . As popt depends on β, a data-driven version is plugged into , with the resulting estimate being asymptotically as good as . Simulations demonstrate the superiority of the convex combination over the Pickands estimate already for moderate sample sizes n.

On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions

We establish the functional central limit theorem and law of the iterated logarithm for the Pickands estimator of the dependence function of a bivariate extreme-value distribution. Our methods are based on general results for sums of random variables taking values in Banach spaces.