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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]

In this paper, we revisit the construction of confidence intervals for extreme quantiles of Pareto‐type distributions. A novel asymptotic pivotal quantity is proposed for these quantile estimators, which leads to new asymptotic confidence intervals that exhibit more accurate coverage probability. This pivotal quantity also allows for the construction of a saddle‐point approximation, from which a second set of new confidence intervals follows. The small‐sample properties and utility of these confidence intervals are studied using simulations and a case study from insurance. [ABSTRACT FROM AUTHOR]

In the late 1960s J.H. Venter started to investigate the use of test statistics for normality based on quadratic distances between the order statistics of the sample and the corresponding hypothetical quantiles. These types of statistics are closely related to the (already known at that time) Shapiro-Wilkes'statistics, for which the limiting distribution was not yet known. The behaviour of the statistics investigated by Venter was such that the standard approach of that time, the so-called stochastic process approach, was insufficient to derive their limiting distribution. However, by writing the statistic in terms of order statistics from a uniform distribution and employing the representation of such order statistics in terms of independent, identically distributed exponential random variables, he was able to approximate the statistic by a quadratic form in independent, identically distributed random variables. This led him to the study of the limiting behaviour of the latter, for which results were not available to handle the statistics he was interested in. The results needed were derived and applied to the statistics of interest, constituting pioneering research that in later years led to the derivation of the limiting distribution of inter alia, the Shapiro-Wilk statistic and many other statistics of a quadratic type. The words of Del Barrio, Cuesta-Albertos, Matran and Rodriguez-Rodriguez in a recent paper, "All the proofs of the asymptotic behaviour of these statistics.., rely on the results in ... "', emphasize the fundamental contributions of Venter's earlier work. In the current paper the above-mentioned contribution and the work flowing from it, are discussed and placed in a historical context. In particular, it is shown that by using an expression for the distribution of uniform order statistics in terms of ratios of sums of independent, identically distributed exponential random variables, the test statistic can be shown to be asymptotically equivalent to a quadratic form in independent, identically distributed random variables. For the latter the results known at that time were insufficient and stronger results had to be developed in order to obtain a limiting distribution. This limiting distribution was obtained as that of a linear combination of independent chi-squared random variables with one degree of freedom each. The latter's characteristic function could be found quite easily and inverted numerically to obtain critical values for the test. The constants in the linear combination are closely related to the eigenvalues of the matrix whose entries are the constants in the quadratic form. It was shown how these constants can be found by transforming the required integral equation into a differential equation for which the classical orthogonal polynomials provide solutions. In this paper it is also shown how these quadratic statistics are related to degenerate U-statistics and correlation-type test statistics, leading to the same limiting distribution. A number of extensions of other researchers are also discussed, as well as more recent developments based on the original work of Venter. To be more specific, at the time of Venter's research, the general approach to deriving the limiting distribution of such quadratic statistics was by means of a so-called stochastic process approach. In that case the statistic was written as a quadratic functional of the empirical process or of the quantile process, for which the limiting distribution was known in terms of the Brownian Bridge process. However, that approach could not be applied to the type of statistics Venter considered, which led him to consider the approach based on quadratic forms discussed above. During the past 30+ years the available tools on empirical and quantile processes have developed immensely, which meant that the stochastic process approach was again a viable option.… [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]