Rank-based inference for bivariate extreme-value copulas

Consider a continuous random pair $(X,Y)$ whose dependence is characterized by an extreme-value copula with Pickands dependence function $A$. When the marginal distributions of $X$ and $Y$ are known, several consistent estimators of $A$ are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and Cap\'{e}ra\`{a}, Foug\`{e}res and Genest [Biometrika 84 (1997) 567--577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of $X$ and $Y$ are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given.Comment: Published in at http://dx.doi.org/10.1214/08-AOS672 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Conversation with Martin Bradbury Wilk

Martin Bradbury Wilk was born on December 18, 1922, in Montr\'{e}al, Qu\'{e}bec, Canada. He completed a B.Eng. degree in Chemical Engineering in 1945 at McGill University and worked as a Research Engineer on the Atomic Energy Project for the National Research Council of Canada from 1945 to 1950. He then went to Iowa State College, where he completed a M.Sc. and a Ph.D. degree in Statistics in 1953 and 1955, respectively. After a one-year post-doc with John Tukey, he became Assistant Director of the Statistical Techniques Research Group at Princeton University in 1956--1957, and then served as Professor and Director of Research in Statistics at Rutgers University from 1959 to 1963. In parallel, he also had a 14-year career at Bell Laboratories, Murray Hill, New Jersey. From 1956 to 1969, he was in turn Member of Technical Staff, Head of the Statistical Models and Methods Research Department, and Statistical Director in Management Sciences Research. He wrote a number of influential papers in statistical methodology during that period, notably testing procedures for normality (the Shapiro--Wilk statistic) and probability plotting techniques for multivariate data. In 1970, Martin moved into higher management levels of the American Telephone and Telegraph (AT&T) Company. He occupied various positions culminating as Assistant Vice-President and Director of Corporate Planning. In 1980, he returned to Canada and became the first professional statistician to serve as Chief Statistician. His accomplishments at Statistics Canada were numerous and contributed to a resurgence of the institution's international standing. He played a crucial role in the reinstatement of the Cabinet-cancelled 1986 Census.Comment: Published in at http://dx.doi.org/10.1214/08-STS272 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the covariance of the asymptotic empirical copula process

Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. Illustrations are provided and consequences for inference are outlined.Comment: 14 pages, 2 figure

Maty's Biography of Abraham De Moivre, Translated, Annotated and Augmented

November 27, 2004, marked the 250th anniversary of the death of Abraham De Moivre, best known in statistical circles for his famous large-sample approximation to the binomial distribution, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability theory. He also made seminal contributions in analytic geometry, complex analysis and the theory of annuities. The first biography of De Moivre, on which almost all subsequent ones have since relied, was written in French by Matthew Maty. It was published in 1755 in the Journal britannique. The authors provide here, for the first time, a complete translation into English of Maty's biography of De Moivre. New material, much of it taken from modern sources, is given in footnotes, along with numerous annotations designed to provide additional clarity to Maty's biography for contemporary readers.Comment: Published at http://dx.doi.org/10.1214/088342306000000268 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Conversation with Herbert Tate: Mathematics Educator and Builder

Herbert Tate was a Professor of Mathematics at McGill University (Montréal, Canada) from 1921 to 1964. As the author of four textbooks, and in his capacity as Chairman of the Department of Mathematics from 1948 to 1960, he played a key role in structuring the institution’s research and study programs in mathematics during an important period of growth. McGill’s current position as a hub of mathematical research owes much to him. In this interview given shortly after his retirement, Herbert Tate describes his career and shares some of his views about mathematics and related topics. Beyond its archival value, this interview reminds us of the extent to which infrastructures and mentalities have changed, at least in Canada, over the past century

Generalized Logistic Models and its orthant tail dependence

The Multivariate Extreme Value distributions have shown their usefulness in environmental studies, financial and insurance mathematics. The Logistic or Gumbel-Hougaard distribution is one of the oldest multivariate extreme value models and it has been extended to asymmetric models. In this paper we introduce generalized logistic multivariate distributions. Our tools are mixtures of copulas and stable mixing variables, extending approaches in Tawn (1990), Joe and Hu (1996) and Foug\`eres et al. (2009). The parametric family of multivariate extreme value distributions considered presents a flexible dependence structure and we compute for it the multivariate tail dependence coefficients considered in Li (2009)

A Primer on Copulas for Count Data

The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete on

La perception du risque de titres financiers : l’importance relative et l’influence de certains facteurs de risque

Cet article présente une analyse de l’importance relative et de l’influence de différents facteurs socio-économiques, institutionnels et systémiques sur la perception du risque de titres financiers chez des gestionnaires de portefeuilles. Les résultats proviennent d’une enquête menée en juin 1990 auprès d’une vingtaine d’investisseurs institutionnels québécois auxquels on avait demandé de quantifier l’importance relative qu’ils accordent à sept facteurs généralement considérés comme représentatifs de l’ensemble des facettes du risque et adéquats pour la détermination du classement du risque de titres provenant de deux secteurs de l’économie, celui des banques et celui de la consommation. Les préférences des répondants, exprimées à partir de comparaisons par paires, ont été étudiées à l’aide d’une variante statistique, proposée par De Jong (1984), du procédé d’analyse hiérarchique de Saaty. Il ressort de cette enquête qu’en dépit d’une grande diversité d’opinions quant à l’importance relative à accorder aux différents facteurs de risque, le classement des titres d’un même secteur varie très peu d’un facteur à l’autre. Une explication de ce phénomène est suggérée.This paper presents an analysis of the relative importance and influence of various socio-economic, institutional and systemic factors considered relevant to portfolio managers' risk perception of securities. The analysis is based on data from a 1990 survey in which twenty institutional investors from Québec were asked to assess the relative importance of seven factors generally regarded as representative of the multidimensional nature of risk and deemed adequate for determining the respondents' risk rankings of securities from the banking and consumer goods sectors. The individuals' preferences were elicited through paired comparisons expressed on a 1-9 scale and analyzed using a statistical variant, developed by De Jong (1984), of Saaty's Analytic Hierarchy Process. Despite the wide range of opinions observed as to the relative importance of the various risk factors, the risk rankings of the securities considered turned out to be very similar across factors. A possible explanation of this phenomenon is offered