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 Extreme Value Index

Summary This paper investigates 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 sufficient, sometimes also necessary, condition, under which the oracle property holds. Some key words: Extreme value index, Distributed inference, Distributed Hill estimator

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.

A Different Approach for Choosing a Threshold in Peaks over Threshold

In Extreme Value methodology the choice of threshold plays an important role in efficient modelling of observations exceeding the threshold. The threshold must be chosen high enough to ensure an unbiased extreme value index but choosing the threshold too high results in uncontrolled variances. This paper investigates a generalized model that can assist in the choice of optimal threshold values in the γ positive domain. A Bayesian approach is considered by deriving a posterior distribution for the unknown generalized parameter. Using the properties of the posterior distribution allows for a method to choose an optimal threshold without visual inspection.

Extreme Value Index Estimation by Means of an Inequality Curve

A characterizing property of Zenga (1984) inequality curve is exploited in order to develop an estimator for the extreme value index of a distribution with regularly varying tail. The approach proposed here has a nice graphical interpretation which provides a powerful method for the analysis of the tail of a distribution. The properties of the proposed estimation strategy are analysed theoretically and by means of simulations. The usefulness of the method will be tested also on real data sets.

On the Maximum Likelihood Estimation of Extreme Value Index Based on k-Record Values

In this paper, we study the existence and consistency of the maximum likelihood estimator of the extreme value index based on k-record values. Following the method used by Drees et al. (2004) and Zhou (2009), we prove that the likelihood equations, in terms of k-record values, eventually admit a strongly consistent solution without any restriction on the extreme value index, which is not the case in the aforementioned studies.

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

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.