Sandra Mendonça

Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology

In this article, we deal with the estimation, under a semi-parametric framework, of a positive extreme value index γ, the primary parameter in Statistics of Extremes, and associated estimation of the Value at Risk (VaR) at a level p, the size of the loss occurred with a small probability p. We consider second-order minimum-variance reduced-bias (MVRB) estimators, and propose the use of bootstrap computer-intensive methods for the adaptive choice of thresholds, both for γ and Var p . Applications in the fields of insurance and finance, as well as a small-scale simulation study of the bootstrap adaptive estimators’ behaviour, are also provided.

Multiple intersection local times in terms of white noise

We show how multiple intersections of Brownian motion can be expressed in terms of generalized white noise functionals. We also calculate the kernels of their chaos expansions and discuss their L2 properties.

Higher-order Monotone Functions and Probability Theory

From topological properties of the cone of real functions with higher order monotonicity properties, we derive an extreme point Choquet representation theorem; this, in fact, follows from Krein-Milman’s theory, since there is a natural compact convex base. This fills the gap between convexity and complete monotonicity, in the sense that we obtain integral representations for functions such that (-l) k f (k) (x) ≥ 0, k = 0,1,…, n (in the form of special “beta transforms”), that in the case of completely monotone functions is Bernstein’s classical result about Laplace transforms of non-decreasing functions. The integral representation is used to reobtain Khinchine’s characterization of unimodal distribution functions and Polya’s characteristic functions. On the other hand, we obtain extended classes of unimodal distributions (different from the star-shaped unimodality of Olshen and Savage), and put in a proper light Mejzler’s classes in extreme value theory, and special classes of characteristic functions such as Sakovic’s class. Using fractional calculus, we have straightforward extensions of the previous results, and consider generalized beta transforms, naturally associated with order statistics with general parent distribution.

Population Genetics of Ochlerotatus eatoni (Diptera: Culicidae) Endemic Species to Two Macaronesian Islands

Abstract Analyses of 11 isoenzyme loci of Ochlerotatus eatoni (Edwards, 1916), endemic to two Macaronesian Islands (Madeira and Tenerife, Canary Islands), revealed substantial genetic structure in the study populations. Samples from sites on the south and north of Madeira displayed a significant reduction of variability compared with those from central Madeira and Tenerife. The Tenerife population exhibited a severe deficit of heterozygosity with similar magnitude across all the loci examined. The complex pattern of variation in Oc. eatoni is because of interplay of breeding structure, genetic drift, and geographical and historical factors. From these findings, we concluded that island colonization by Oc. eatoni was not marked by founder effect.

Randomly Stopped {\varvec{k}}th Order Statistics

Randomly stopped order statistics when the stopping rule is generated by a basic count distribution are investigated. Unified expressions in terms of the subordinator are presented, extending results from geometrically thinned sequences. Using the results on limit stable distributions for max-geometric laws, and Smirnov’s techniques to deal with limit laws of extreme order statistics, some results on stability of Panjer subordinated randomly stopped order statistics are discussed.

Application of the Theory of Extremes to the Study of Precipitation in Madeira Island: Statistical Choice of Extreme Domains of Attraction

In the past and nowadays, hydrology is one of the most natural fields of application for the theory of extremes. This work presents an application of univariate extreme value theory to the study of precipitation in Madeira Island. The method for testing extreme value conditions investigated by Dietrich et al. (Extremes 5:71–85, 2002) was applied to the monthly 1-day maxima precipitation data for the rainy season from seven pluviometric stations maintained by the Portuguese Meteorological Institute. The statistical procedures for the problem of statistical choice of extreme domains of attraction analysed by Neves and Fraga Alves (TEST 16:297–313, 2007) were also applied to each station data set. The results of this analysis indicate the possible k upper extremes to be used for each local sample and the sign of each extreme value index γ.