Lígia Henriques-Rodrigues

The field of statistical extreme value theory (EVT) focuses on estimating parameters associated with extreme events, such as the probability of exceeding a high threshold or determining a high quantile that lies at or beyond the observed data range. Typically, the assumption for univariate data analysis is that the sample is complete, independent, identically distributed, or weakly dependent and stationary, drawn from an unknown distribution F. However, in the context of lifetime data, censoring is a common issue. In this work, we consider the case of random censoring for data with a heavy-tailed, Pareto-type distribution. As is common in applications of EVT, the estimation of the extreme value index (EVI) is critical, as it quantifies the tail heaviness of the distribution. The EVI has been extensively studied in the literature. Here, we discuss several classical EVI-estimators and reduced-bias (RB) EVI-estimators within a semi-parametric framework, with a focus on RB EVI-estimators derived from generalized means, which will be applied to both simulated and real survival data.

The estimation of the extreme value index (EVI) is a crucial task in the field of statistics of extremes, as it provides valuable insights into the tail behavior of a distribution. For models with a Pareto-type tail, the Hill estimator is a popular choice. However, this estimator is susceptible to bias, which can lead to inaccurate estimations of the EVI, impacting the reliability of risk assessments and decision-making processes. This paper introduces a novel reduced-bias generalized Hill estimator, which aims to enhance the accuracy of EVI estimation by mitigating the bias.

The Weibull tail‐coefficient (WTC) plays a crucial role in extreme value statistics when dealing with Weibull‐type tails. Several distributions, such as normal, Gamma, Weibull, and logistic distributions, exhibit this type of tail behavior. The WTC, denoted by , is a parameter in a right‐tail function of the form , where represents a regularly varying cumulative hazard function with an index of regular variation equal to 1/ . The commonly used WTC‐estimators in literature are often defined as functions of the log‐excesses, making them closely related to estimators of the extreme value index (EVI) for Pareto‐type tails. For a positive EVI, the classical estimator is the Hill estimator. Generalized means have been successfully employed in estimating the EVI, leading to reduction of bias and of root mean square error for appropriate threshold values. In this study, we propose and investigate new classes of WTC‐estimators based on power of the log‐excesses within a second‐order framework. The performance of these new estimators is evaluated through a large‐scale Monte Carlo simulation study, comparing them with existing WTC‐estimators available in the literature.

Due to the specificity of the Weibull tail coefficient, most of the estimators available in the literature are based on the log excesses and are consequently quite similar to the estimators used for the estimation of a positive extreme value index. The interesting performance of estimators based on generalized means leads us to base the estimation of the Weibull tail coefficient on the power mean-of-order-. Consistency and asymptotic normality of the estimators under study are put forward. Their performance for finite samples is illustrated through a Monte Carlo simulation. It is always possible to find a negative value of (contrarily to what happens with the mean-of-order- estimator for the extreme value index), such that, for adequate values of the threshold, there is a reduction in both bias and root mean square error.