Probability With a View Towards Statistics, Two Volume Set

Description

Volume I of this two-volume text and reference work begins by providing a foundation in measure and integration theory. It then offers a systematic introduction to probability theory, and in particular, those parts that are used in statistics. This volume discusses the law of large numbers for independent and non-independent random variables, transforms, special distributions, convergence in law, the central limit theorem for normal and infinitely divisible laws, conditional expectations and martingales. Unusual topics include the uniqueness and convergence theorem for general transforms with characteristic functions, Laplace transforms, moment transforms and generating functions as special examples. The text contains substantive applications, e.g., epidemic models, the ballot problem, stock market models and water reservoir models, and discussion of the historical background. The exercise sets contain a variety of problems ranging from simple exercises to extensions of the theory. Volume II of this two-volume text and reference work concentrates on the applications of probability theory to statistics, e.g., the art of calculating densities of complicated transformations of random vectors, exponential models, consistency of maximum estimators, and asymptotic normality of maximum estimators. It also discusses topics of a pure probabilistic nature, such as stochastic processes, regular conditional probabilities, strong Markov chains, random walks, and optimal stopping strategies in random games. Unusual topics include the transformation theory of densities using Hausdorff measures, the consistency theory using the upper definition function, and the asymptotic normality of maximum estimators using twice stochastic differentiability. With an emphasis on applications to statistics, this is a continuation of the first volume, though it may be used independently of that book. Assuming a knowledge of linear algebra and analysis, as well as a course in modern probability, Volume II looks at statistics from a probabilistic point of view, touching only slightly on the practical computation aspects.

J. Hoffman-Jorgensen (University of Aarhus, Denmark) (Author)

..."an exceptionally thorough and scholarly treatise on modern probability and selected major topics in mathematical statistics. The volumes provide an exciting and potentially profitable opportunity, both for reading and teaching...a fresh and exciting opportunity for researchers, instructors and students a like...the high quality of the main material made it difficult for me to put the books down long enough to complete this review." -Publication of the International Statistical Institute ..."a useful text for library shelves and reference." -Journal of the Royal Statistical Society ..."This is one of the few monographs which combines a deep and rigorous treatment of measure theory on the one hand with applications on the other hand." -Zbl. Math.