Dinis Pestana

A Sturdy Reduced-Bias Extreme Quantile (VaR) Estimator

Please see the supplementary material for an important correction. The main objective of statistics of extremes lies in the estimation of quantities related to extreme events. In many areas of application, such as statistical quality control, insurance, and finance, a typical requirement is to estimate a high quantile, that is, the value at risk at a level p (VaRp), high enough so that the chance of an exceedance of that value is equal to p, small. In this article we deal with the semiparametric estimation of VaRp for heavy tails. The classical semiparametric estimators of parameters characterizing the tail behavior of the underlying model F usually exhibit a high bias for low thresholds, that is, for large values of k, the number of top order statistics used for the estimation. We shall here deal with bias reduction techniques for heavy tails, trying to improve the performance of the classical high quantile estimators through the use of an adequate bias-corrected tail index estimator. The new high quantile estimators have a mean squared error smaller than that of the classical estimators, even for small values of k. They are, thus, alternatives to the classical estimators not only around optimal levels but also for other levels. The asymptotic distributional properties of the proposed classes of estimators are derived. The estimators are compared with alternative ones, not only asymptotically but also for finite samples, through Monte Carlo techniques. An application to the analysis of different datasets in the field of finance is also provided.

A simple second-order reduced bias’ tail index estimator

In this article, we are interested in the direct estimation of the dominant component of the bias of a classical tail index estimator, such as the Hill estimator, used here for illustration of the procedure. Such an estimated bias is then directly removed from the original estimator. The second-order parameters in the bias are based on a number of top order statistics, larger than the one we should use for the estimation of the tail index γ, so that there is no change in the asymptotic variance of the new reduced bias’ tail index estimator, which is kept equal to the asymptotic variance of the classical original one, contrarily to what happens with most of the reduced bias’ estimators available in the literature. The asymptotic distributional behaviour of the proposed estimators of γ is derived, under a second-order framework, and their finite sample properties are also obtained through Monte Carlo simulation techniques.

Modeling carcass removal time for avian mortality assessment in wind farms using survival analysis

In monitoring studies at wind farms, the estimation of bird and bat mortality caused by collision must take into account carcass removal by scavengers or decomposition. In this paper we propose the use of survival analysis techniques to model the time of carcass removal. The proposed method is applied to data collected in ten Portuguese wind farms. We present and compare results obtained from semiparametric and parametric models assuming four main competing lifetime distributions (exponential, Weibull, log-logistic and log-normal). Both homogeneous parametric models and accelerated failure time models were used. The fitted models enabled the estimation of the carcass persistence rates and the calculation of a scavenging correction factor for avian mortality estimation. Additionally, we discuss the impact that the distributional assumption can have on parameter estimation. The proposed methodology integrates the survival probability estimation problem with the analysis of covariate effects. Estimation is based on the most suitable model while simultaneously accounting for censored observations, diminishing scavenging rate estimation bias. Additionally, the method establishes a standardized statistical procedure for the analysis of carcass removal time in subsequent studies.

A simple generalisation of the Hill estimator

The classical Hill estimator of a positive extreme value index (EVI) can be regarded as the logarithm of the geometric mean, or equivalently the logarithm of the mean of order p = 0 , of a set of adequate statistics. A simple generalisation of the Hill estimator is now proposed, considering a more general mean of order p ? 0 of the same statistics. Apart from the derivation of the asymptotic behaviour of this new class of EVI-estimators, an asymptotic comparison, at optimal levels, of the members of such class and other known EVI-estimators is undertaken. An algorithm for an adaptive estimation of the tuning parameters under play is also provided. A large-scale simulation study and an application to simulated and real data are developed.

Tail index and second-order parameters’ semi-parametric estimation based on the log-excesses

In this paper we are interested in the derivation of the asymptotic and finite-sample distributional properties of a ‘quasi-maximum likelihood’ estimator of a ‘scale’ second-order parameter β, directly based on the log-excesses of an available sample. Such estimation is of primordial importance for the adaptive selection of the optimal sample fraction to be used in the classical semi-parametric tail index estimation as well as for the reduced-bias estimation of the tail index, high quantiles and other parameters of extreme or even rare events. An application in the area of survival analysis is provided, on the basis of a data set on males diagnosed with cancer of the tongue.

Populational Growth Models Proportional to Beta Densities with Allee Effect

We consider populations growth models with Allee effect, proportional to beta densities with shape parameters p and 2, where the dynamical complexity is related with the Malthusian parameter r. For p>2, these models exhibit a population dynamics with natural Allee effect. However, in the case of 1<p⩽2, the proposed models do not include this effect. In order to inforce it, we present some alternative models and investigate their dynamics, presenting some important results.

Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology

In this article, we deal with the estimation, under a semi-parametric framework, of a positive extreme value index γ, the primary parameter in Statistics of Extremes, and associated estimation of the Value at Risk (VaR) at a level p, the size of the loss occurred with a small probability p. We consider second-order minimum-variance reduced-bias (MVRB) estimators, and propose the use of bootstrap computer-intensive methods for the adaptive choice of thresholds, both for γ and Var p . Applications in the fields of insurance and finance, as well as a small-scale simulation study of the bootstrap adaptive estimators’ behaviour, are also provided.

Populational growth models in the light of symbolic dynamics

Using symbolic dynamic techniques, populational growth models proportional to beta densities, are investigated. Our results give explicit methods to investigate the chaotic behaviour of populational growth models, when the malthusean parameter increases. The chaotic complexity is measured in terms of the topological entropy.

Extensions of Dorfman’s Theory

Economic impact of composite sampling is investigated in the realistic framework of tests with positive probability of false positive and of false negative results. Sensitivity and specificity when pooling samples are also discussed, using rarefaction as a framework.

New Reduced-bias Estimators of a Positive Extreme Value Index

Noting that the classical Hill estimator of a positive extreme value index (EVI) is the logarithm of the mean of order-0 of a set of certain statistics, a more general class of EVI-estimators based on the mean of order-p (MOP), p ⩾ 0, of such statistics was recently introduced. The asymptotic behavior of the class of MOP EVI-estimators is reviewed, and compared to their reduced-bias MOP (RBMOP) and optimal RBMOP versions, which are suggested here and studied both asymptotically and for finite samples, through a large-scale simulation study. Applications to simulated datasets are also put forward.