M. Fátima Brilhante

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.

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.

A new partially reduced-bias mean-of-order p class of extreme value index estimators

A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order- p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k , the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.

The MOP EVI-Estimator Revisited

A simple generalisation of the classical Hill estimator of a positive extreme value index (EVI) has been recently introduced in the literature. Indeed, the Hill estimator 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. Instead of such a geometric mean, it is thus sensible to consider the mean of order p (MOP) of those statistics, with p ≥ 0. In this paper, a small-scale simulation study and a closer look at the asymptotic behaviour at optimal levels of the class of MOP EVI-estimators enable us to better understand their properties and to suggest simple adaptive EVI-estimates.

On the proportion of non uniform reported p-values

The family Xm, m ∈ [−2, 2], whose density is obtained tilting the uniform density using the pole (0.5, 1) has as extreme cases the Beta(1, 2) density, corresponding to m=−2, and the Beta(2, 1) density, corresponding to m=2. For intermediate m, the density mixes the uniform density (m=0) with one of the extreme cases. It is therefore an appropriate model for reported p-values when the best, or worst, of two replica is recorded whenever the result of the experiment is unexpected. Estimation of the mixing parameter is discussed.

Further results on order statistics and products of functions of independent generalized beta random variables

The BetaBoop(p, q, P, Q), p, q, P, Q > 0, family of random variables includes as special cases Beta(p, q) random variables, which are known to be quite versatile in modeling different randomness patterns. Some results of the multiplicative algebra of BetaBoop random variables are commented in the light of uniform order statistics. Products and products of powers and of random powers of independent standard uniforms are also investigated, namely when the exponent is either uniform or the maximum of an uniform random sample, or its probability density function is a mixture of the uniform density and the density of an extreme of two independent uniform random variables.

A Mean-of-Order-\(p\) Class of Value-at-Risk Estimators

The main objective of statistics of univariate extremes lies in the estimation of quantities related to extreme events. In many areas of application, like finance, insurance and statistical quality control, a typical requirement is to estimate a high quantile, i.e. the Value at Risk at a level \(q (\)VaR\(_q)\), high enough, so that the chance of exceedance of that value is equal to \(q\), with \(q\) small. In this paper we deal with the semi-parametric estimation of VaR\(_q\), for heavy tails, introducing a new class of VaR-estimators based on a class of mean-of-order- \(p\) (MOP) extreme value index (EVI)-estimators, recently introduced in the literature. Interestingly, the MOP EVI-estimators can have a mean square error smaller than that of the classical EVI-estimators, even for small values of \(k\). They are thus a nice basis to build alternative VaR-estimators not only around optimal levels, but for other levels too.The new VaR-estimators are compared with the classical ones, not only asymptotically, but also for finite samples, through Monte-Carlo techniques.

Cantor Sets with Random Repair

The effect of random repair in each step of the construction of Cantor-like sets, defined by the union of segments determined by the minimum and maximum of two independent observations from a population with support on [0, 1], is investigated here. Independence between the samples used in the damage and repair stages is also assumed. The final assessment of the repair benefits is done in terms of the mean diameter and mean total length of the set obtained after a small number of iterations.

A non-parametric double-bootstrap method for an adaptive MOP EVI-estimation

The Hill estimator, the average of k excesses of ordered log-observations, can be regarded as the logarithm of the mean of order p = 0 of a set of adequate statistics. The mean of order p (MOP), now with p ≥ 0, of the same statistics leads to the so-called MOP extreme value index (EVI)-estimator, a simple generalisation of the classical Hill estimator of a positive EVI, recently introduced in the literature. This class of MOP EVI-estimators depends on the extra tuning parameter p ≥ 0, which makes it very flexible, and even able to overpass most of the ‘classical’ and even reduced-bias EVI-estimators. Apart from a simulation study that reflects such an assertion, we advance with a fully non-parametric double bootstrap algorithm for the choice of p and k. We further provide applications of the algorithm to simulated and real data in the fields of biostatistics.