Frederico Caeiro

In Statistics of Extremes, the tail shape second order parameter is a relevant parameter whenever we want to improve the estimation of first order parameters. We shall consider two semi-parametric estimators of the shape second order parameter, parameterized with a tuning parameter. We provide a Monte Carlo comparative simulation study of several algorithms for the choice of such tuning parameter and for an adaptive estimation of the shape second order parameter.In Statistics of Extremes, the tail shape second order parameter is a relevant parameter whenever we want to improve the estimation of first order parameters. We shall consider two semi-parametric estimators of the shape second order parameter, parameterized with a tuning parameter. We provide a Monte Carlo comparative simulation study of several algorithms for the choice of such tuning parameter and for an adaptive estimation of the shape second order parameter.

In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an application to real data.In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an application to real data.

For heavy tails, classical extreme value index estimators, like the Hill estimator, are usually asymptotically biased. Consequently those estimators are quite sensitive to the number of top order statistics used in the estimation. The recent minimum-variance reduced-bias extreme value index estimators enable us to remove the dominant component of asymptotic bias and keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator. In this paper a new minimum-variance reduced-bias extreme value index estimator is introduced, and its non degenerate asymptotic behaviour is studied. A comparison with another important minimum-variance reduced-bias extreme value index estimator is also provided.For heavy tails, classical extreme value index estimators, like the Hill estimator, are usually asymptotically biased. Consequently those estimators are quite sensitive to the number of top order statistics used in the estimation. The recent minimum-variance reduced-bias extreme value index estimators enable us to remove the dominant component of asymptotic bias and keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator. In this paper a new minimum-variance reduced-bias extreme value index estimator is introduced, and its non degenerate asymptotic behaviour is studied. A comparison with another important minimum-variance reduced-bias extreme value index estimator is also provided.

n/a

Floods are recurrent events that can have a catastrophic impact. In this work we are interested in the analysis of a data set of gauged daily flows from the Whiteadder Water river, Scotland. Using statistic techniques based on extreme value theory, we estimate several extreme value parameters, including extreme quantiles and return periods of high levels.Floods are recurrent events that can have a catastrophic impact. In this work we are interested in the analysis of a data set of gauged daily flows from the Whiteadder Water river, Scotland. Using statistic techniques based on extreme value theory, we estimate several extreme value parameters, including extreme quantiles and return periods of high levels.

The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the extreme value index (EVI). Whenever we are interested in large values, such estimation is usually performed on the basis of the largest k + 1 order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive EVI-estimation. Such a choice can be either heuristic or based on sample paths stability or on the minimization of a mean squared error estimateas a function of k. Some of these procedures will be reviewed. Despite of thefact that the methods provided can be applied, with adequate modifications, to any real EVI and not only to the adaptive EVI-estimation but also to the adaptive estimation of other relevant right-tail parameters, we shall illustrate the methods essentially for the EVI and for heavy tails, i.e., for a positive EVI.The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the extreme value index (EVI). Whenever we are interested in large values, such estimation is usually performed on the basis of the largest k + 1 order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive EVI-estimation. Such a choice can be either heuristic or based on sample paths stability or on the minimization of a mean squared error estimateas a function of k. Some of these procedures will be reviewed. Despite of thefact that the methods provided can be applied, with adequate modifications, to any real EVI and not only to the adaptive EVI-estimation but also to the adaptive estimation of other relevant right-tail parameters, we shall illustrate the methods essentially for the EVI and for heavy tails, i.e., for a positive EVI.

n/a

n/a

n/a

{Summary: We consider a new class of consistent semi-parametric estimators of a negative extreme value index, based on the set of the $k$ largest observations. This class of estimators depends on a control or tuning parameter, which enables us to have access to an estimator with a null second-order component of asymptotic bias, and with a rather interesting mean squared error, as a function of $k$. We study the consistency and asymptotic normality of the proposed estimators. Their finite sample behaviour is obtained through Monte Carlo simulation.}

In extreme value theory, the shape second-order parameter is a quite relevant parameter related to the speed of convergence of maximum values, linearly normalized, towards its limit law. The adequate estimation of this parameter is vital for improving the estimation of the extreme value index, the primary parameter in statistics of extremes. In this article, we consider a recent class of semi-parametric estimators of the shape second-order parameter for heavy right-tailed models. These estimators, based on the largest order statistics, depend on a real tuning parameter, which makes them highly flexible and possibly unbiased for several underlying models. In this article, we are interested in the adaptive choice of such tuning parameter and the number of top order statistics used in the estimation procedure. The performance of the methodology for the adaptive choice of parameters is evaluated through a Monte Carlo simulation study.In extreme value theory, the shape second-order parameter is a quite relevant parameter related to the speed of convergence of maximum values, linearly normalized, towards its limit law. The adequate estimation of this parameter is vital for improving the estimation of the extreme value index, the primary parameter in statistics of extremes. In this article, we consider a recent class of semi-parametric estimators of the shape second-order parameter for heavy right-tailed models. These estimators, based on the largest order statistics, depend on a real tuning parameter, which makes them highly flexible and possibly unbiased for several underlying models. In this article, we are interested in the adaptive choice of such tuning parameter and the number of top order statistics used in the estimation procedure. The performance of the methodology for the adaptive choice of parameters is evaluated through a Monte Carlo simulation study.

{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 first consider a class of consistent semi-parametric estimators of a positive tail index $\gamma$, parametrised in a tuning or control parameter $\alpha$. Such a control parameter enables us to have access for any available sample, to an estimator of the tail index $\gamma$ with a null dominant component of asymptotic bias and consequently with a reasonably flat mean squared error pattern, as a function of $k$, the number of top-order statistics considered.\par Such a control parameter depends on a second-order parameter $\rho$, which will be adequately estimated so that we may 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 use a number of top-order statistics larger than the one usually required for the estimation through the Hill estimator. An illustration of the behaviour of the estimators is provided, through the analysis of the daily log-returns on the Euro-US\$ exchange rates.}