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In this paper, we introduce a robust estimator of the tail index of a Pareto-type distribution. The estimator is obtained through the use of the minimum density power divergence with an exponential regression model for log-spacings of top order statistics. The proposed estimator is compared to existing minimum density power divergence estimators of the tail index based on fitting an extended Pareto distribution and exponential regression model on log-ratio of spacings of order statistics. We derive the influence function and gross error sensitivity of the proposed estimator of the tail index to study its robustness properties. In addition, a simulation study is conducted to assess the performance of the estimators under different contaminated samples from different distributions. The results show that our proposed estimator of the tail index has better mean square errors and is less sensitive to an increase in the number of top order statistics. In addition, the estimation of the exponential regression model yields estimates of second-order parameters that can be used for estimation of extreme events such as quantiles and exceedance probabilities. The proposed estimator is illustrated with practical datasets on insurance claims and calcium content in soil samples. [ABSTRACT FROM AUTHOR]

This article proposes an open-set recognition model that is based on the use of extreme value statistics. For this purpose, a distance ratio is introduced that expresses how dissimilar a target point is from the known classes by considering the ratio of distances locally around the target point. It is shown that the class of generalized Pareto distributions with bounded support can be used to model the peaks of the distance ratio above a high threshold. The resulting distribution provides a probabilistic framework to perform open-set recognition. Furthermore, we describe a numerical procedure to estimate the hyperparameters of our model. This procedure is based on a new objective function that considers both the fit of the generalized Pareto distribution and the misclassification error of the known classes. Our method is applied to three image datasets and an audio dataset showing that it outperforms similar open-set recognition and anomaly detection methods. for this article are available online. [ABSTRACT FROM AUTHOR]

Regression Quantiles (RQs) (see Koenker and Bassett, 1978) can be found as optimal solutions to a Linear Programming (LP) problem. Also, these optimal solutions correspond to specific elemental regressions (ERs). On the other hand, single case ordinary least squares (OLS) leverage statistics can be expressed as weighted averages of ER ones. Using this three-tier relationship amongst RQs, ERs, and OLS leverage statistics some relationships between single case leverage statistics and ER ones are explored and deduced. We build upon these results and propose a multiple-case RQ weighted predictive leverage statistic,TJ. We do this using an ER view of the well-known leverage relationship,, by summing the ER weighted predictive leverage statistics over all ERs (RQs included) instead of over observations, i.e.,. As an ad-hoc cut-off value of this statistic we make use of the analog of the Hoaglin and Welsch (1978) one, i.e., high leverage points have. So in the RQ weighted predictive leverage scenario, the cut-off value becomes, whereKis the total number of ERs. We then apply this RQ high leverage diagnostic to well-known data sets in the literature. The cut-off value used generally seems too small. Some proposals of cut-off values based on some analytical bounds and a simulation study are therefore given and shown to be reasonable. [ABSTRACT FROM PUBLISHER]

We develop and study in the framework of Pareto-type distributions a class of nonparametric kernel estimators for the conditional second order tail parameter. The estimators are obtained by local estimation of the conditional second order parameter using a moving window approach. Asymptotic normality of the proposed class of kernel estimators is proven under some suitable conditions on the kernel function and the conditional tail quantile function. The nonparametric estimators for the second order parameter are subsequently used to obtain a class of bias-corrected kernel estimators for the conditional tail index. In particular it is shown how for a given kernel function one obtains a bias-corrected kernel function, and that replacing the second order parameter in the latter with a consistent estimator does not change the limiting distribution of the bias-corrected estimator for the conditional tail index. The finite sample behavior of some specific estimators is illustrated with a simulation experiment. The developed methodology is also illustrated on fire insurance claim data. [ABSTRACT FROM AUTHOR]

We introduce new families of estimators for the Weibull-tail coefficient, obtained from a weighted sum of a power transformation of excesses over a high random threshold. Asymptotic normality of the estimators is proven for an intermediate sequence of upper order statistics, and under classical regularity conditions for L-statistics and a second-order condition on the tail behavior of the underlying distribution. The small sample performance of two specific examples of kernel functions is evaluated in a simulation study. [ABSTRACT FROM AUTHOR]

The estimation of extreme conditional quantiles is an important issue in different scientific disciplines. Up to now, the extreme value literature focused mainly on estimation procedures based on independent and identically distributed samples. Our contribution is a two-step procedure for estimating extreme conditional quantiles. In a first step nonextreme conditional quantiles are estimated nonparametrically using a local version of [Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica , 46 , 33-50.] regression quantile methodology. Next, these nonparametric quantile estimates are used as analogues of univariate order statistics in procedures for extreme quantile estimation. The performance of the method is evaluated for both heavy tailed distributions and distributions with a finite right endpoint using a small sample simulation study. A bootstrap procedure is developed to guide in the selection of an optimal local bandwidth. Finally the procedure is illustrated in two case studies. [ABSTRACT FROM AUTHOR]

In this paper we assess a marking scheme for the test for logical order in English language testing. This is achieved by considering the potential misclassification of students due to guessing. The distribution of scores under the hypothesis of random selection is derived and the asymptotic distribution is shown to be Poisson with a mean of unity. Recommendations are given for selecting the number of sentences in the test for logical order. [ABSTRACT FROM PUBLISHER]