Inference and Asymptotics
Our book Asymptotic Techniquesfor Use in Statistics was originally planned as an account of asymptotic statistical theory, but by the time we had completed the mathematical preliminaries it seemed best to publish these separately. The present book, although largely self-contained, takes up the original theme and gives a systematic account of some recent developments in asymptotic parametric inference from a likelihood-based perspective. Chapters 1-4 are relatively elementary and provide first a review of key concepts such as likelihood, sufficiency, conditionality, ancillarity, exponential families and transformation models. Then first-order asymptotic theory is set out, followed by a discussion of the need for higher-order theory. This is then developed in some generality in Chapters 5-8. A final chapter deals briefly with some more specialized issues. The discussion emphasizes concepts and techniques rather than precise mathematical verifications with full attention to regularity conditions and, especially in the less technical chapters, draws quite heavily on illustrative examples. Each chapter ends with outline further results and exercises and with bibliographic notes. Many parts of the field discussed in this book are undergoing rapid further development, and in those parts the book therefore in some respects has more the flavour of a progress report than an exposition of a largely completed theory.
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Some general concepts
preliminaries and motivations
Some tools of higherorder theory
ancillary statistic apply approximation asymptotically normal Barndorff-Nielsen Barndorff-Nielsen and Cox Bartlett adjustment calculation canonical parameter canonical statistic censoring chi-squared components conditional distribution consider corresponding covariance matrix cumulants defined degrees of freedom denote depend discussion equivalent error Example expected information exponential distribution exponential family follows formula given identically distributed independent and identically inference information matrix inverse Gaussian distribution likelihood function likelihood ratio statistic location-scale log-likelihood function marginal likelihood maximum likelihood estimate minimal sufficient modified profile likelihood normal distribution normalizing constant notation Note nuisance parameters null hypothesis observed information obtained order O(n orthogonal p”-formula parameter of interest parameter space parameterization particular Poisson problems profile log-likelihood properties random variables regression repeated sampling respectively results and exercises score function score statistic score vector Show ſº specification standard normal sufficient statistic Suppose transformation models unknown parameter variance
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