Frederico Caeiro

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.

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.

{Summary: For heavy-tailed models, and working with the sample of the $k$ largest observations, we present probability weighted moments (PWM) estimators for the first order tail parameters. Under regular variation conditions on the right-tail of the underlying distribution function $F$ we prove the consistency and asymptotic normality of these estimators. Their performance, for finite sample sizes, is illustrated through a small-scale Monte Carlo simulation.}

{Summary: In many areas of application, like, for instance, climatology, hydrology, insurance, finance, and statistical quality control, a typical requirement is to estimate a high quantile of probability $1 - p$, a value high enough so that the chance of an exceedance of that value is equal to $p$, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index $\gamma $, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for $\gamma >0$, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles' estimators, with the same asymptotic variance as the (biased) ``classical'' estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.}

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In this article, we revisit Feuerverger and Halls maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.In this article, we revisit Feuerverger and Halls maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.

{Summary: We are interested in the comparison, under a third-order framework, of classes of second-order, reduced-bias tail index estimators, giving particular emphasis to minimum-variance reduced-bias estimators of the tail index $\gamma$. The full asymptotic distributional properties of the proposed classes are derived under a third-order framework and the estimators are compared with other alternatives, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the USA Dollar 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.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.

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A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order-p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k, the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order-p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k, the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.

{Let $X_i$ be i.i.d. r.v.s with heavy-tailed CDF $F(x)$ such that $$1-F(x)=(x/C)^{-1/\gamma}((1+(\beta/\rho)(x/C)^{\rho/\gamma} +\beta'(x/C)^{2\rho/\gamma}(1+o(1))),$$ where $\gamma$ is the tail index ($\gamma>0$), and $\rho<0$ and $\beta$ are the ``second order parameters''. The authors construct an estimator for $\beta$ based on the ``tail moments'' $$M_n^{(\alpha)}=(k)^{-1}\sum_{i=1}^k [łog X_{n-i+1:n}-łog X_{n-k:n}]^\alpha. $$ Consistency and asymptotic normality of the estimator are demonstrated. The small sample properties of the estimator are investigated via simulations.}

{Summary: Heavy tailed-models are quite useful in many fields, like insurance, finance, telecommunications, internet traffic, among others, and it is often necessary to estimate a high quantile, i.e., a value that is exceeded with a probability $p$, small. The semiparametric estimation of this parameter relies essentially on the estimation of the tail index, the primary parameter in statistics of extremes. Classical semi-parametric estimators of extreme parameters show usually a severe bias and are known to be very sensitive to the number $k$ of top order statistics used in the estimation. For $k$ small they have a high variance, and for large $k$ a high bias. Recently, new second-order ``shape'' and ``scale'' estimators allowed the development of second-order reduced-bias estimators, which are much less sensitive to the choice of $k$. Here we study, under a third order framework, minimum-variance reduced-bias (MVRB) tail index estimators, recently introduced in the literature, and dependent on an adequate estimation of second order parameters. The improvement comes from the asymptotic variance, which is kept equal to the asymptotic variance of the classical Hill estimator [ıt B. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] provided that we estimate the second order parameters at a level of a larger order than the level used for the estimation of the first order parameter. The use of those MVRB tail index estimators enables us to introduce new classes of reduced-bias high quantile estimators. These new classes are compared among themselves and with previous ones through the use of a small-scale Monte Carlo simulation.}

Reduced-bias versions of a very simple generalization of the ‘classical’ Hill estimator of a positive extreme value index (EVI) are put forward. The Hill estimator can be regarded as the logarithm of the mean-of-order-0 of a certain set of statistics. Instead of such a geometric mean, it is sensible to consider the mean-of-order-p (MOP) of those statistics, with p real. Under a third-order framework, the asymptotic behaviour of the MOP, optimal MOP and associated reduced-bias classes of EVI-estimators is derived. Information on the dominant non-null asymptotic bias is also provided so that we can deal with an asymptotic comparison at optimal levels of some of those classes. Large-scale Monte-Carlo simulation experiments are undertaken to provide finite sample comparisons.Reduced-bias versions of a very simple generalization of the ‘classical’ Hill estimator of a positive extreme value index (EVI) are put forward. The Hill estimator can be regarded as the logarithm of the mean-of-order-0 of a certain set of statistics. Instead of such a geometric mean, it is sensible to consider the mean-of-order-p (MOP) of those statistics, with p real. Under a third-order framework, the asymptotic behaviour of the MOP, optimal MOP and associated reduced-bias classes of EVI-estimators is derived. Information on the dominant non-null asymptotic bias is also provided so that we can deal with an asymptotic comparison at optimal levels of some of those classes. Large-scale Monte-Carlo simulation experiments are undertaken to provide finite sample comparisons.