Mathematical statistics

A wide-ranging, extensive overview of modern mathematical statistics, this work reflects the current state of the field while being succinct and easy to grasp. The mathematical presentation is coherent and rigorous throughout. The author presents classical results and methods that form the basis of modern statistics, and examines the foundations of estimation theory, hypothesis testing theory and statistical game theory. He then considers statistical problems for two or more samples, and those in which observations are taken from different distributions. Methods of finding optimal and asymptotically optimal statistical procedures are given, along with treatments of homogeneity testing, regression, variance analysis and pattern recognition. The author also posits a number of methodological improvements that simplify proofs, and brings together a number of new results which have never before been published in a single monograph. In Sections 1-4, we introduce the notions of a sample and an empirical distribution, and then study their basic properties, mostly asymptotic ones, which constitute the basic of mathematical statistics.