Asymptotics for the Hirsch Index

The last decade methods for quantifying the research output of individual researchers have become quite popular in academic policy making. The h- index (Hirsch, 2005) constitutes an interesting quality measure that has attracted a lot of attention recently. It is now a standard measure available for instance on theWeb of Science. In this paper we establish the asymptotic normality of the empirical h-index. The rate of convergence is non-standard: ph=(1 + nf(h)), where f is the density of the citation distribution and n the number of publications of a researcher. In case that the citations follow a Pareto-type or a Weibull-type distribution as defined in extreme value theory, our general result nicely specializes to results that are useful for constructing confidence intervals for the h-index.Asymptotic normality;confidence interval;extreme value theory;research output;scientometrics;tail empirical process.

Unbiased Tail Estimation by an Extension of the Generalized Pareto Distribution

AMS classifications: 62G20; 62G32;bias;exchange rate;heavy tails;peaks-over-threshold;regular variation;tail index

Mandelbrot's Extremism

In the sixties Mandelbrot already showed that extreme price swings are more likely than some of us think or incorporate in our models.A modern toolbox for analyzing such rare events can be found in the field of extreme value theory.At the core of extreme value theory lies the modelling of maxima over large blocks of observations and of excesses over high thresholds.The general validity of these models makes them suitable for out-of-sample extrapolation.By way of illustration we assess the likeliness of the crash of the Dow Jones on October 19, 1987, a loss that was more than twice as large as on any other single day from 1954 until 2004.exceedances;extreme value theory;heavy tails;maxima

Semiparametric Lower Bounds for Tail Index Estimation

indexation;semiparametric estimation

Estimating the maximum possible earthquake magnitude using extreme value methodology: the Groningen case

The area-characteristic, maximum possible earthquake magnitude $T_M$ is required by the earthquake engineering community, disaster management agencies and the insurance industry. The Gutenberg-Richter law predicts that earthquake magnitudes $M$ follow a truncated exponential distribution. In the geophysical literature several estimation procedures were proposed, see for instance Kijko and Singh (Acta Geophys., 2011) and the references therein. Estimation of $T_M$ is of course an extreme value problem to which the classical methods for endpoint estimation could be applied. We argue that recent methods on truncated tails at high levels (Beirlant et al., Extremes, 2016; Electron. J. Stat., 2017) constitute a more appropriate setting for this estimation problem. We present upper confidence bounds to quantify uncertainty of the point estimates. We also compare methods from the extreme value and geophysical literature through simulations. Finally, the different methods are applied to the magnitude data for the earthquakes induced by gas extraction in the Groningen province of the Netherlands

Rapid variation with remainder and rates of convergence

(op voorblad per abuis E. Beirland vermeld, in plaats van J. Beirlant) The remainder term of the class \Gamma$ of rapidly varying fuctions Is considered. Some probabilistic applIcations to limit laws of extreme value theory and to the estlmation of the indexparameter of a regularly varying tail are considered. Keywords and Phrases: regular variation, rates of convergence, domains of attraction

Bahadur-Kiefer theorems for the product-limit process

AbstractIn the random censorship from the right model, strong and weak limit theorems for Bahadur-Kiefer type processes based on the product-limit estimator are established. The main theorm is sharp and may be considered as a final result as far as this type of research is concerned. As a consequence of this theorem a sharp uniform Bahadur representation for product-limit quantiles is obtained

Asymptotic confidence intervals for the length of the shortt under random censoring

A shortt of a one dimensional probability distribution is defined to be an interval which has at least probability t and minimal length. The length of a shortt and its obvious estimator are significant measures of scale of a distribution and the corresponding random sample. respectively. In this note a non-parametric asymptotic confidence interval for the length of the (uniqueness is assumed) shortt is established in the random censorship from the right model. The estimator of the length of the shortt is based on the prOduct-limit (PL) estimator of the unknown distribution function. The proof of the result mainly follows from an appropriate combination of the Glivenko·Cantelli theorem and the functional central limit theorem for the PL estimator

Asymptotic confidence intervals for the length of the Shortt under random censoring

A shortt of a one dimensional probability distribution is defined to be an interval which has at least probability t and millimallength. The length of a shortt and its obvious estimator are significant measures of scale of a distribution and the corresponding random sample, respectively. In this note a non-parametric asymptotic confidence interval for the length of the (uniqueness is assumed) shortt is established in the random censorship from the right model. The estimator of the length of the shortt is based on the product-limit (PL) estimator of the unknown distribution function. The proof of the result mainly follows from an appropriate combination of the Glivenko-Cantelli theorem and the functional central limit theorem for the PL estimator. Key words and phrases: Confidence interval, length of shortt, random censorship