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Improvements in the estimation of the Weibull tail coefficient -- a comparative study

Lígia Henriques-Rodrigues, Frederico Caeiro, M. Ivette Gomes

Abstract

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 behaviour. The WTC, denoted by $θ$, is a parameter in a right-tail function of the form $ \bar F(x) :=1-F(x) =: {\rm e}^{-H(x)}$, where $H(x)=-\ln(1-F(x))$ represents a regularly varying cumulative hazard function with an index of regular variation equal to 1/$θ$, $θ\in\mathbb{R}^{+}$. 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 $p$ 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.

Improvements in the estimation of the Weibull tail coefficient -- a comparative study

Abstract

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 behaviour. 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.
Paper Structure (10 sections, 10 theorems, 75 equations, 6 figures)

This paper contains 10 sections, 10 theorems, 75 equations, 6 figures.

Table of Contents

  1. Introduction and scope of the article
  2. Semi-parametric EVI- and WTC-estimators
  3. Asymptotic non-degenerate behaviour
  4. Asymptotic behaviour of the WTC-estimators under study
  5. Asymptotic MSE of the estimators
  6. Finite sample properties of the WTC-estimators
  7. Mean values and RMSE patterns as $k$-functionals
  8. Mean value and RMSE at optimal simulated levels
  9. Conclusions
  10. Simulated optimal sample fraction

Key Result

theorem 1

Under a second-order framework, i.e., assuming that condition SOC holds, with $k$ an intermediate sequence, as in intermediate, the WTC-estimator $\widehat{\theta}^{\rm GG}_p(k)$, in WTC-Mop-GG, has an asymptotic distributional representation of the type, with $P_k$ a standard normal RV, frequently denoted by $\mathcal{N}(0,1)$.

Figures (6)

  • Figure 1: Simulated mean values (left) and RMSE (right) for the Exponential model.
  • Figure 2: Simulated mean values (left) and RMSE (right) for the Weibull(2,1) model.
  • Figure 3: Simulated mean values (left) and RMSE (right) for the Gamma(0.75,1) model.
  • Figure 4: Simulated mean values (left) and RMSE (right) for the Half-Normal model.
  • Figure 5: Simulated mean values (left) and RMSE (right) for the standard Gumbel model.
  • ...and 1 more figures

Theorems & Definitions (18)

  • theorem 1: Caeiro et al.caeiroetal-wtcmop
  • theorem 2
  • proof
  • corollary 1
  • proof
  • corollary 2
  • corollary 3
  • theorem 3
  • proof
  • theorem 4
  • ...and 8 more