{Consider the distribution function $EV_{\gamma}(x)=\exp(-(1+\gamma x)^{- 1/\gamma}),\ \gamma>0,\ 1+\gamma x>0$, to which $\max(X_{1},łdots, X_{n})$ is attracted after suitable linear normalization. The authors consider the underlying model $F$ in the max-domain of attraction of $EV_{\gamma}$, where $ X_{i:n},\ 1łeq Iłeq n$, denotes the i-th ascending order statistic associated to the random sample $(X_{1},łdots, X_{n})$ from the unknown distribution function $F$. This article is devoted to studying semi-parametric estimators of $\gamma$ in the form $$\gamma_{n}^{(þeta,\alpha)}(k)=(\Gamma(\alpha)/M_{n}^{(\alpha- 1)}(k))łeft(M_{n}^{(þeta\alpha)}(k)/\Gamma(þeta\alpha+1)\right) ^{1/þeta},\quad \alpha\geq 1,\quad þeta>0,$$ parametrized by the parameters $\alpha$ and $þeta$, which may be controlled, where $M_{n}^{(0)}=1$ and $ M_{n}^{(\alpha)}(k)=k^{-1}\sum_{i=1}^{k}(łn X_{n-i+1:n}-łn X_{n-k:n})^{\alpha}$, $\alpha>0$, is a consistent estimator of $\Gamma(\alpha+1)\gamma^{\alpha}$, as $k\toınfty$, and $k=o(n)$, as $n\toınfty$.\par The authors derive the asymptotic distributional properties of the considered class of estimators and obtain that for $þeta>1$ it is always possible to find a control parameter $\alpha$ which makes the dominant component of the asymptotic bias of the proposed estimator null and depends on the second order parameter $\rho$. An investigation of the $\rho$-estimator is presented.}

{Summary: We consider a class of consistent semi-parametric estimators of a positive tail index $\gamma$, parameterized by a tuning or control parameter $\alpha$. Such a control parameter enables us to have access, for any available sample, to an estimator of $\gamma$ with a null dominant component of asymptotic bias, and with a reasonably flat mean squared error pattern, as a function of $k$, the number of top order statistics considered. Moreover, we are able to achieve a high efficiency relative to the classical Hill estimator [ıt B. M. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)], provided we may have access to a larger number of top order statistics than the number needed for optimal estimation through the Hill estimator.}

Plasma is the fourth state of matter, following the three states of solid, liquid and gas. Experience has amply demonstrated that solids exposed to the oxygen-deficient plasma flame are converted to liquid, and liquid exposed to the same flame is converted to gas. A low amount of vitrified solid residue material usually remains at the end of this process. Plasma pyrolysis/vitrification (PP/V) has been demonstrated as a safe, efficient, cost-effective technology for the treatment of wastes, including hazardous wastes. Besides the low amounts of gaseous byproducts that PP/V produces, the solid vitrified residue presents a low leachability of pollutants. Studies have been performed in many countries in order to assess the leachability of chemical substances. But from the results of identified studies, none has reported results on the ecotoxicological properties of the leachates. The aim of this study was to contribute to the assessment of ecotoxic risk of four different vitrified materials. Vitrified samples of contaminated soils, municipal solid wastes, and incinerator bottom ashes were submitted to the European leaching pre-standard test number prEN 12457-2. The leachates were analyzed for 22 chemical parameters. The biological characterization comprised the assessment of bioluminescence inhibition of Photobacterium phosphoreum bacterium, growth inhibition of Pseudokirchneriella subcapitata algae and the germination inhibition of Lactuca sativa vegetable. The chemical and ecotoxicological results were analyzed according to the French proposal of Criteria on the Evaluation Methods of Waste Toxicity (CEMWT) and a Toxicity Classification System (TCS). The chemical and ecotoxicological results indicated a low leachability of pollutants and a low toxicity level of leachates. All samples studied were as below the TCS class 1 level (no significant toxicity observed) and as non-ecotoxic for CEMWT. Therefore, the environmental ecotoxic risk of the analyzed vitrified samples was determined to be very low.

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