Principles of Statistical Inference: From a Neo-Fisherian Perspective
In this book, an integrated introduction to statistical inference is provided from a frequentist likelihood-based viewpoint. Classical results are presented together with recent developments, largely built upon ideas due to R.A. Fisher. The term “neo-Fisherian” highlights this.After a unified review of background material (statistical models, likelihood, data and model reduction, first-order asymptotics) and inference in the presence of nuisance parameters (including pseudo-likelihoods), a self-contained introduction is given to exponential families, exponential dispersion models, generalized linear models, and group families. Finally, basic results of higher-order asymptotics are introduced (index notation, asymptotic expansions for statistics and distributions, and major applications to likelihood inference).The emphasis is more on general concepts and methods than on regularity conditions. Many examples are given for specific statistical models. Each chapter is supplemented with problems and bibliographic notes. This volume can serve as a textbook in intermediate-level undergraduate and postgraduate courses in statistical inference.
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DATA AND MODEL REDUCTION
SURVEY OF SOME BASIC CONCEPTS
EXPONENTIAL DISPERSION FAMILIES
ASYMPTOTIC EXPANSIONS FOR STATISTICS
ASYMPTOTIC EXPANSIONS FOR DISTRIBUTIONS
ancillary statistic asymptotic distribution asymptotic expansion Barndorff-Nielsen and Cox Bartlett correction components composite group family conditional likelihood confidence regions Consider corresponding covariance matrix cumulant generating function defined denote depend distribution constant statistic distribution function distribution Let Edgeworth expansion element equivariant error of order evaluated Example exponential families expression finite formula Furthermore gamma distribution given group Q Hence independent observations indices inverse likelihood function likelihood quantities linear models location family log-likelihood ratio marginal likelihood maximal invariant maximum likelihood estimate minimal sufficient statistic model reduction natural exponential family natural parameter normal distribution notation nuisance parameter null distribution null expectation obtain one-parameter one-to-one order O(n~l orthogonal parameter of interest parameterization invariant po(y Poisson probability problem profile likelihood pseudo-likelihood random sampling random variables reparameterization respect scalar scale and location score sequence Show statistical model sufficient statistic term of order uniformly most powerful usually variance vector zero