M. João Martins

“Asymptotically Unbiased” Estimators of the Tail Index Based on External Estimation of the Second Order Parameter

In this paper we shall deal with the asymptotic and finite sample properties of “asymptotically unbiased” estimators of the tail index γ, based on “external” adequate estimators of the second order parameter ρ. The behavior of the ρ-estimator considered has indeed a high impact on the distributional properties of the final estimator of γ, and must be carefully chosen. As a by-product of the final study we present also the finite sample properties of a few ρ-estimators available in the literature.

Generalizations of the Hill estimator – asymptotic versus finite sample behaviour☆

Abstract The main goal of this paper is to present generalized Hill estimators parametrized in a positive real α (and equal to the Hill estimator when α =1), which are asymptotically more efficient than the Hill estimator for a large region of values of α for any point of the ( γ , ρ )-plane, where γ >0 is the tail index , related to the heaviness of the tail 1− F of the underlying model F , and ρ ⩽0 is the second-order parameter , related to the rate of convergence of maximum values, linearly normalized, towards its limit. The practical validation of asymptotic results for small finite samples is done by means of simulation techniques in Frechet and Burr models, and some indications are provided on the choice of α .

Bias reduction and explicit semi-parametric estimation of the tail index☆

In this paper, and in a context of regularly varying tails, we analyse particular but interesting cases of the maximum likelihood and least squares estimators proposed by Feuerverger and Hall (Ann. Statist. 27 (1999) 760). All these estimators are alternatives to a well-known estimator of the tail index, the Hill estimator (Ann. Statist. 3 (1997) 1163), and jointly with the generalized jackknife estimators in Gomes et al. (Extremes 2 (2000) 207, Portug. Math. 59 (2002) 393) have essentially in mind a reduction in bias, preferably without increasing mean squared error, leading to semi-parametric estimators of the tail index with a better performance than the classical estimators, provided we may use extreme-value data relatively deep into the sample.

Generalized Jackknife-Based Estimators for Univariate Extreme-Value Modeling

In this article, we revisit the importance of the generalized jackknife in the construction of reliable semi-parametric estimates of some parameters of extreme or even rare events. The generalized jackknife statistic is applied to a minimum-variance reduced-bias estimator of a positive extreme value index—a primary parameter in statistics of extremes. A couple of refinements are proposed and a simulation study shows that these are able to achieve a lower mean square error. A real data illustration is also provided.

Some results on the behaviour of hill's estimator

The maximum of a large number of random variables, suitably normalized, follows, under certain general conditions, the Generalized Extreme Value (GEV) distribution . Hill’s estimator provides an estimation for γ > 0 from a finite sample, requiring the largest m observations, out of n. For different underlying distributions with heavy tails, we provide some exact results on the bias of Hill’s estimator, simulation results on the dependence on m of the mean squared error (MSE), bias and variance of such estimator and a comparison of simulation and asymptotic results. Additionally, we present some work on convex combinations of Hill’s estimator. The simulated mixture relative efficiency, based on MSE, although not high, shows the existence of some possible improvement.

A Combinatorial Approach to Assess the Separability of Clusters

Separability of clusters is an issue that arises in many different areas, and is often used in a rather vague and subjective manner. We introduce a combinatorial notion of interiority to derive a global view on separability of a set of entities. We develop this approach further to evaluate the overall separability of a partition in the context of cluster analysis. Our approach captures combinatorial and geometrical aspects of data and provides, in addition to numerical evaluations, graphical representations particularly useful when data are not easily visualized. We illustrate the methodology on some real and simulated datasets.

Averages of Hill estimators

Averaging Hills estimators leads to a reduction in the volatility of Hills plot. We deal with a generalization of the procedure proposed by Resnick and Stărică (1997), and, propose alternatives, assymptotically equivalent at the respective optimal levels, but with more interesting sample paths. Asymptotic normality is derived for intermediate levels where the asymptotic bias may be non-null. A simulation study completes the asymptotic results and shows the advantages of the proposed estimators in the problem of choosing the number of the top order statistics to be used in the estimation of the tail index.

Data depth for the uniform distribution

Given a set

Generalized jackknife semi-parametric estimators of the tail index.

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