Jozef L. Teugels

Tail index estimation, Pareto quantile plots, and regression diagnostics

Abstract Successful application of extreme value statistics for estimating the Pareto tail index relies heavily on the choice of the number of extreme values taken into account. It is shown that these tail index estimators can be considered estimates of the slope at the right upper tail of a Pareto quantile plot, obtained using a weighted least squares algorithm. From this viewpoint, based on classical ideas on regression diagnostics, algorithms can be constructed searching for that order statistic to the right of which one obtains an optimal linear fit of the quantile plot.

Markov Chains: Models, Algorithms and Applications

Teugels reviews Markov Chains: Models, Algorithms and Applications by Wai-Ki Ching and Michael K. Ng.

The Politics of Large Numbers: a History of Statistical Reasoning

This book’s main theme is the gradual but distinct development of statistics as a scientix8e c discipline and as an administrative activity over the last three centuries. The author does this by constructing a genealogical tree that combines the interplay between econometrics and inferential statistics. Side players in this development are mathematics, astronomy, politics, and—to an even larger extent—biometrics. Chapter 1 presents the emergence of administrative statistics in Germany, France, and England in the early parts of the eighteenth century. Chapter 2 deals with the partly simultaneous and earliest achievements of probability calculus, in particular measurement questions in astronomy, the law of large numbers, the x8e rst central limit theorem, and the method of least squares. Chapter 3 focuses on averages, and treats the ideas and efforts of Quetelet in some detail. The concepts of correlation and regression as they emanate from the investigations of Galton and Pearson are covered in Chapter 4. In Chapters 5 and 6 the author suggests a few of the specix8e c ties among bureaus of statistics in different states, the structures of particular states, and other objects of social analysis for the period that lasted from the 1830s (when many of these bureaus were created) to the 1940s. Four countries are compared: France and Great Britain, where the unix8e ed state was an ancient and legitimate phenomenon, and Germany and the United States, countries that experienced gestation or rapid growth. Chapter 7 provides a treatment of the social conditions under which sampling techniques originated. The issue of how to collect a representative sample from a given population started with the Norwegian Kiaer to get its formal solution, and stratix8e ed sampling was presented by Neyman. Chapter 8 deals with taxonomic questions and is illustrated by examples as natural species, branches of industry, poverty, unemployment, social categories, and causes of death. The x8e nal chapter explores the difx8e culties involved in uniting the four traditions leading to modern econometrics: economic theory, descriptive historicist statistics, mathematical statistics resulting from biometrics, and probability calculus. Overall, Desrosières deals with the interaction between the rise and development of statistical ideas and methods on the one hand, and the way these methods have been used by governments on the other hand. Understandably, the author draws much of his material from French sources. The role played by French scientists in the early development of probability and statistics has indeed been substantial. Moreover, the centralizing role of the government has also been clear. From a historical standpoint, this book covers the period from roughly 1700 to 1950. It is a thoughtful, basically philosophical book, with no new ground broken in history. Readers interested in the history of statistics, economics, and sociometry will x8e nd pleasure in this sophisticated study.

Simultaneous Success-Run Chains

If every element from a source of identical particles has to perform a certain fixed success-run Markov chain with n + 2 states, and if the particles are put into the initial state one at a time, they act independently and reach the absorbing state n + 2 after n + 2 steps or they return to the source. This sequence of simultaneous success-run chains can be analysed by using a basic Markov chain with 2n different states that is studied in great detail. Among others, we derive the transition matrix and the stationary distribution. It further turns out that, for m âx89§ n the basic Markov chain already behaves in a stationary way, so that it is easy to find the distribution of the number of absorbed particles at a certain instant, as well as the number of particles that return to the source. The proofs of the main theorems are based on mathematical induction and matrix methods.