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Inference and Asymptotics
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Likelihood and its many associated concepts are of central importance in statistical theory and applications. The theory of likelihood and of likelihood-like objects (pseudo-likelihoods) has undergone extensive and important developments over the past 10 to 15 years, in particular as regards higher order asymptotics. This book provides an account of this field, which is still vigorously expanding. Conditioning and ancillarity underlie the p*-formula, a key formula for the conditional density of the maximum likelihood estimator, given an ancillary statistic. Various types of pseudo-likelihood are discussed, including profile and partial likelihoods. Special emphasis is given to modified profile likelihood and modified directed likelihood, and their intimate connection with the p*-formula. Among the other concepts and tools employed are sufficiency, parameter orthogonality, invariance, stochastic expansions and saddlepoint approximations. Brief reviews are given of the most important properties of exponential and transformation models and these types of model are used as test-beds for the general asymptotic theory. A final chapter briefly discusses a number of more general issues, including prediction and randomization theory. The emphasis is on ideas and methods, and detailed mathematical developments are largely omitted. There are numerous notes and exercises, many indicating substantial further results.
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Table of Contents

Introduction Preliminaries Some general concepts First order theory Higher order theory:preliminaries Mathematical basis of higher order theory Higher order theory: likelihood combinants higher order theory: some further results and tools Various notions of likelihood and higher order theory Further aspects References Index

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"The book succeeds in bringing together a large amount of useful and interesting work." -Short Book Reviews "The book is carefully structured. The presentation is generally graded so that motivations, concepts, and results are first discussed, at least briefly, with a minimum of technicalities, and then more complete details are given...usually illustrated with several examples. This style is very helpful and allows the casual reader to grasp the purposes and flow of the developments without requiring that every derivation be followed." -Journal of the American Statistical Association "This is an authoritative and comprehensive book which develops its complex theoretical results in a coherent and logical manner. Anybody interested in recent developments in parametric likelihood theory should read it." -The Statistician

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