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62F12 Statistics -- Parametric inference -- Asymptotic properties of estimators

62A01 Statistics -- Foundational and philosophical topics -- Foundations and philosophical topics En-ligne: OA - libre accès

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One of the categories into which may be classified asymptotic results given in the literature relating to parametric statistical inference is the following: the model is restricted to a certain class of “well- behaved” models and then the result is proved strictly under a few extra specified conditions. It is the purpose of the monograph under review to extend the applicability of this approach by introducing a class of statistical models (the analytic models) which is sufficiently well behaved to satisfy regularity conditions of the type typically met in theorems of asymptotic statistical inference, and at the same time sufficiently rich to contain many of the commonly used statistical models, including the curved exponential families.

It is demonstrated in this monograph that the class of analytic models (a.m.’s) provides a suitable framework for further development of asymptotic theory of likelihood based inference for parametric statistical models. The author hopes “that by use of this framework it will become easier to provide rigorous proofs of results of second and higher order statistical inference, without confining the results to curved exponential models”. The monograph consists of five chapters.

Chapter 1 contains auxiliary mathematical concepts and results, in particular in relation to multilinear functions. Chapter 2 begins with the definition of the class of a.m.’s and continues to explore its basic properties. Some basic auxiliary results of a.m.’s are derived. The index of an a.m. is defined and its behavior in connection with independent observations is investigated. Possibilities of obtaining analytic models from other analytic models are explored. These include analytic reparametrizations, and reductions by sufficiency and ancillarity. Chapter 3 contains examples of a.m.’s and classes of such. It is demonstrated here that the a.m.’s contain the sufficiently smooth curved exponential families as well as models that are not of this type.

Chapters 4 and 5 contain examples of applications to asymptotic statistical theory. These are either examples of first order asymptotic results for general sequences of models (Chapter 4) or of higher-order asymptotic results for models for independent replications (Chapter 5). Some open problems and possible lines of further research are listed at the end of the monograph. The bibliography contains 28 items. (Zentralblatt)

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