@article {1022.62022,
	title = {Bias reduction in the estimation of parameters of rare events.},
	journal = {Theory of Stochastic Processes},
	volume = {8},
	year = {2002},
	note = {<p>n/a</p>
},
	pages = {67-76},
	abstract = {<p>{Consider the distribution function $EV_{\gamma}(x)=\exp(-(1+\gamma x)^{- 1/\gamma}),\ \gamma\&gt;0,\ 1+\gamma x\&gt;0$, to which $\max(X_{1},{\l}dots, X_{n})$ 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},\ 1{\l}eq I{\l}eq n$, denotes the i-th ascending order statistic associated to the random sample $(X_{1},{\l}dots, X_{n})$ from the unknown distribution function $F$. This article is devoted to studying semi-parametric estimators of $\gamma$ in the form $$\gamma_{n}^{({\th}eta,\alpha)}(k)=(\Gamma(\alpha)/M_{n}^{(\alpha- 1)}(k)){\l}eft(M_{n}^{({\th}eta\alpha)}(k)/\Gamma({\th}eta\alpha+1)\right) ^{1/{\th}eta},\quad \alpha\geq 1,\quad {\th}eta\&gt;0,$$ parametrized by the parameters $\alpha$ and ${\th}eta$, which may be controlled, where $M_{n}^{(0)}=1$ and $ M_{n}^{(\alpha)}(k)=k^{-1}\sum_{i=1}^{k}({\l}n X_{n-i+1:n}-{\l}n X_{n-k:n})^{\alpha}$, $\alpha\&gt;0$, is a consistent estimator of $\Gamma(\alpha+1)\gamma^{\alpha}$, as $k\to{\i}nfty$, and $k=o(n)$, as $n\to{\i}nfty$.\par The authors derive the asymptotic distributional properties of the considered class of estimators and obtain that for ${\th}eta\&gt;1$ 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.}</p>
},
	keywords = {estimators, heavy tails}, rare events, {bias reduction},
	attachments = {https://docentes.fct.unl.pt/sites/default/files/fac/files/2002tsp_caeiro_gomes.pdf},
	author = {Caeiro, Frederico and Gomes, M. Ivette}
}
