Measuring heavy-tailedness of distributions (original) (raw)

Tails and probabilities for extreme outliers

The task of estimation of the tails of probability distributions having small samples seems to be still opened and almost unsolvable. The paper tries to make a step in filling this gap. In 2017 Jordanova et al. introduce six new characteristics of the heaviness of the tails of theoretical distributions. They rely on the probability to observe mild or extreme outliers. The main their advantage is that they always exist. This work presents some new properties of these characteristics. Using them six distribution sensitive estimators of the extremal index are defined. A brief simulation study compares their quality with the quality of Hill, t-Hill, Pickands and Deckers-Einmahl-de Haan estimators.

Heavy-Tailed Distributions: Data, Diagnostics, and New Developments

SSRN Electronic Journal, 2000

This monograph is written for the numerate nonspecialist, and hopes to serve three purposes. First it gathers mathematical material from diverse but related fields of order statistics, records, extreme value theory, majorization, regular variation and subexponentiality. All of these are relevant for understanding fat tails, but they are not, to our knowledge, brought together in a single source for the target readership. Proofs that give insight are included, but for fussy calculations the reader is referred to the excellent sources referenced in the text. Multivariate extremes are not treated. This allows us to present material spread over hundreds of pages in specialist texts in twenty pages. Chapter 5 develops new material on heavy tail diagnostics and gives more mathematical detail.

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.

A Diagnostic Plot for Estimating the Tail Index of a Distribution

Journal of Computational and Graphical Statistics, 2004

The problem of estimating the tail index in heavy-tailed distributions is very important in many applications. We propose a new graphical method that deals with this problem by selecting an appropriate number of upper order statistics. We also investigate the method's theoretical properties are investigated. Several real datasets are analyzed using this new procedure and a simulation study is carried

SOME INDICES FOR HEAVY-TAILED DISTRIBUTIONS

It is known that, for certain probability distributions, some classical risk-indices, depending on moments on the first and the second order, may not exist. In this paper some alternative indices are presented in order to give a classification of both the most common distributions and the only heavy-tailed (or subexponential) ones.

A test for comparing tail indices for heavy tailed distributions via empirical likelihood

In this work, the problem of testing whether different ($\geq 2$) independent samples, with (possibly) different heavy tailed distributions, share the same extreme value index, is addressed. The test statistic proposed is inspired by the empirical likelihood methodology and consists in an ANOVA-like confrontation of Hill estimators. Asymptotic validity of this simple procedure is proved and efficiency, in terms of empirical type I error and power, is investigated through simulations under a variety of situations. Surprisingly, this topic had hardly been addressed before, and only in the two sample case, though it can prove useful in applications.

Estimation of extreme quantiles from heavy and light tailed distributions

Journal of Statistical Planning and Inference, 2012

In , a new family of distributions is introduced, depending on two parameters τ and θ, which encompasses Pareto-type distributions as well as Weibull tail-distributions. Estimators for θ and extreme quantiles are also proposed, but they both depend on the unknown parameter τ , making them useless in practical situations. In this paper, we propose an estimator of τ which is independent of θ. Plugging our estimator of τ in the two previous ones allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a small simulation study and on a real data set.