Estimation of the Weibull Tail Coefficient Through the Power Mean-of-Order-p

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

“…Since 𝜃 > 0, we need to have 𝛼 > 0 in order to achieve this relation between the two AMSEs. Similarly, from (30), we can conclude that if 𝜃 = 2𝛼 p , the two estimators have the same AMSE and that AMSE…”

“…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: $$$$ {\hat{\theta}}_p^{\mathrm{G}}(k):= \frac{{\mathrm{H}}_p(k)}{\frac{1}{k}{\sum}_{i=1}^k\mathrm{lnln}\left(\left(n+1\right)/i\right)-\mathrm{lnln}\left(\left(n+1\right)/\left(k+1\right)\right)},\kern0.30em 1\le k\le n-1,\kern0.30em p\in \mathrm{\mathbb{R}}, $$$$ $$$$ {\tilde{\theta}}_p^{GG}(k):= \ln \left(n/k\right){\mathrm{PM}}_p(k),\kern0.30em 1\le k\le n-1,\kern0.30em p>-1,\kern0.30em p\ne 0, $$$$ and …”

“…A first approach to the use of GMs in the WTC-estimation was recently performed in Caeiro et al [8,30], where the class of GM EVI-estimators H p , in (13), was plugged into the class of WTC-estimators in (16),…”

“…A first approach to the use of GMs in the WTC‐estimation was recently performed in Caeiro et al [8, 30], where the class of GM EVI‐estimators H $$$ {}_p $$$, in (), was plugged into the class of WTC‐estimators in (), $$$$ {\hat{\theta}}_p^{\mathrm{GG}}(k):= \ln \left(n/k\right){\mathrm{H}}_p(k),\kern0.30em 1\le k\le n-1,\kern0.30em p\in \mathrm{\mathbb{R}} $$$$ with $$$ {\hat{\theta}}_0^{\mathrm{GG}}(k)\equiv {\hat{\theta}}^{GG}(k) $$$ in (). The asymptotic properties of the new WTC‐estimators were derived, and the finite sample behavior was assessed through a Monte Carlo simulation study.…”