Introduction to Statistical Inference (Dover Books on Mathematics)

Undergraduate students with knowledge of elementary calculus will find here an excellent text emphasizing the inferential and decision-making aspects of statistics. Since the concept of probability is fundamental to statistical inference, the first chapter is concerned mainly with the elements of the calculus of probability. The second chapter contains the essential statistical techniques of summarizing the data in a sample prior to making inferences about the population.
The author then addresses the general properties of distributions, their cumulants and cumulant generating functions, with reference to a number of special probability distributions (binomial, Poisson, normal, gamma and beta, chi-square, log-normal). This leads up to the relation between a sample and the population from which it is drawn and the concepts of confidence intervals and fiducial inference.
Chapter Six introduces the principles of testing hypotheses and making decisions with assigned risks of error, including the method of maximum likelihood and the concept of the power of a test. A treatment of different sampling procedures follows, including sequential methods, at which point the usual exact statistical tests on samples from a normal population are discussed. Subsequent chapters offer lucid discussions of variance, bivariate problems, non-linear regression and curve-fitting, multivariate problems, and stochastic processes. A special feature of this book is the inclusion of topics not usually found in regular courses — the gamma and beta functions, Stirling’s approximation, Jacobians, Bernouilli numbers, etc. Proofs are given wherever possible and sets of problems are included at the end of each chapter, arranged according to difficulty.