Estimation problems for distributions with heavy tails (original) (raw)

Semi-parametric estimation for heavy tailed distributions

Extremes, 2010

In this paper, we generalize several works in the extreme value theory for the estimation of the extreme value index and the second order parameter. Weak consistency and asymptotic normality are proven under classical assumptions. Some numerical simulations and computations are also performed to illustrate the finite-sample and the limiting behavior of the estimators.

Asymptotic Expansions of Estimators for the Tail Index with Applications

Scandinavian Journal of Statistics, 1998

We present asymptotic expansions for two well-known estimators of the tail index of a distributionÐthe Hill's estimator and the simpli®ed Pickands' estimator. We then use the expansions to get more accurate interval estimates. Comparisons between the two estimators are also discussed.

Estimation of the tail-index in a conditional location-scale family of heavy-tailed distributions

Dependence Modeling, 2019

We introduce a location-scale model for conditional heavy-tailed distributions when the covariate is deterministic. First, nonparametric estimators of the location and scale functions are introduced. Second, an estimator of the conditional extreme-value index is derived. The asymptotic properties of the estimators are established under mild assumptions and their finite sample properties are illustrated both on simulated and real data.

Tail index estimation for heavy tails: accommodation of bias in the excesses over a high threshold

Extremes, 2008

In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called "maximum likelihood" estimators of the tail index γ . In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar "maximum likelihood" estimator of a positive tail index γ , based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study.

Measuring heavy-tailedness of distributions

Different questions related with analysis of extreme values and outliers arise frequently in practice. To exclude extremal observations and outliers is not a good decision, because they contain important information about the observed distribution. The difficulties with their usage are usually related with the estimation of the tail index in case it exists. There are many measures for the center of the distribution, e.g. mean, mode, median. There are many measures for the variance, asymmetry and kurtosis, but there is no easy characteristic for heavy-tailedness of the observed distribution. Here we propose such a measure, give some examples and explore some of its properties. This allows us to introduce classification of the distributions, with respect to their heavy-tailedness. The idea is to help and navigate practitioners for accurate and easier work in the field of probability distributions. Using the properties of the defined characteristics some distribution sensitive extremal index estimators are proposed and their properties are partially investigated.

Tail inference using extreme U-statistics

2022

Abstract: 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 corresponds with the one of the Hájek projection, the proof goes beyond considering the first term in Hoeffding’s variance decomposition; instead, a growing number of terms needs to be incorporated in the proof. To show the usefulness of extreme U-statistics, we propose a kernel depending on the three highest order statistics leading to an unbiased estimator of the shape parameter of the generalized Pareto distribution. Whe...