The Use of Generalized Means in the Estimation of the Weibull Tail Coefficient

Abstract

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- p . 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 p (contrarily to what happens with the mean-of-order- p 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.

Improvements in the estimation of the Weibull tail coefficient: A comparative study

“…Observing the behavior of the function B(t) that controls the bias, we can consider three types of models: (1) models where B(t) = 0, as the Weibull, which generalizes the standard Exponential model; (2) models with B(t) ∝ t 𝛽 = t −1 , as the Logistic and Gumbel models; and (3) models with B(t) ∝ t 𝛽 ln t = ln t∕t, as the Gaussian and Gamma models. From Remark 1, in Caeiro et al [8], and under EVI-estimation, the EVI, 𝜉, and the second-order shape parameter, 𝜌, are both null corresponding to Gumbel's type right-tails (Λ(x) = exp(− exp(−x))), x ∈ R) that entail a penultimate behavior (see Gomes [9,10] and Gomes & de Haan [11] for further details) of the Fréchet-type if 𝜃 > 1 or the max-Weibull-type if 𝜃 < 1.…”

“…In this work, we are going to extend the previous results in Caeiro et al [8,30], studying the asymptotic and finite sample behavior of the following WTC-estimators:…”

“…In this work, we are going to extend the previous results in Caeiro et al [8, 30], studying the asymptotic and finite sample behavior of the following WTC‐estimators: $$$$ {\hat{\theta}}_p&amp;amp;#x0005E;{\mathrm{G}}(k):&amp;amp;#x0003D; \frac{{\mathrm{H}}_p(k)}{\frac{1}{k}{\sum}_{i&amp;amp;#x0003D;1}&amp;amp;#x0005E;k\mathrm{lnln}\left(\left(n&amp;amp;#x0002B;1\right)/i\right)-\mathrm{lnln}\left(\left(n&amp;amp;#x0002B;1\right)/\left(k&amp;amp;#x0002B;1\right)\right)},\kern0.30em 1\le k\le n-1,\kern0.30em p\in \mathrm{\mathbb{R}}, $$$$ $$$$ {\tilde{\theta}}_p&amp;amp;#x0005E;{GG}(k):&amp;amp;#x0003D; \ln \left(n/k\right){\mathrm{PM}}_p(k),\kern0.30em 1\le k\le n-1,\kern0.30em p&amp;gt;-1,\kern0.30em p\ne 0, $$$$ and …”

“…In this subsection, we present the asymptotic MSE (AMSE) of the WTC-estimators under study, defined in ( 17)- (20), for models where the function B(t), in (8), can assume a generic expression, models with B(t) = 𝛼t −1 , as the Logistic and Gumbel distributions, and models where B(t) can be considered null, that is, B(t) = 0, as the Weibull and the standard Exponential models. With θ• p (k) generically denoting any of the aforementioned WTC-estimators, we can write…”

“…Observing the behavior of the function $$$ B(t) $$$ that controls the bias, we can consider three types of models: (1) models where $$$ B(t)&amp;amp;#x0003D;0 $$$, as the Weibull, which generalizes the standard Exponential model; (2) models with $$$ B(t)\propto {t}&amp;amp;#x0005E;{\beta }&amp;amp;#x0003D;{t}&amp;amp;#x0005E;{-1} $$$, as the Logistic and Gumbel models; and (3) models with $$$ B(t)\propto {t}&amp;amp;#x0005E;{\beta}\ln t&amp;amp;#x0003D;\ln t/t $$$, as the Gaussian and Gamma models. From Remark 1, in Caeiro et al [8], and under EVI‐estimation, the EVI, $$$ \xi $$$, and the second‐order shape parameter, $$$ \rho $$$, are both null corresponding to Gumbel's type right‐tails ( $$$ \Lambda (x)&amp;amp;#x0003D;\exp \left(-\exp \left(-x\right)\right)\Big),x\in \mathrm{\mathbb{R}} $$$) that entail a penultimate behavior (see Gomes [9, 10] and Gomes & de Haan [11] for further details) of the Fréchet‐type if $$$ \theta &amp;gt;1 $$$ or the max‐Weibull‐type if $$$ \theta &amp;lt;1 $$$.…”