Asymptotic statistics. With a view to stochastic processes
type: Item
instance of: scholarly article
title: Asymptotic statistics. With a view to stochastic processes
zbMATH Open document ID: 1306.62013
DOI: 10.1515/9783110250282
published in: De Gruyter Textbook
publication date: 2012-10-04T00:00:00Z
reviewed by: Ken-ichi Yoshihara
Mathematics Subject Classification ID: 62-01
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62M05
Mathematics Subject Classification ID: 62Mxx
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 6090158
zbMATH Keywords: statistical experiment
zbMATH Keywords: contiguity
zbMATH Keywords: log-likelihood ratio
zbMATH Keywords: weak convergence
zbMATH Keywords: Gaussian shift model
zbMATH Keywords: mixed normal experiment
author: R. Höpfner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110250282
OpenAlex ID: W4256040896
review text: This book is devoted to the study of general asymptotic theory of statistical experiments. The core is developed in Chapters 5 (Gaussian shift models), 6 (Quadratic experiments and mixed normal experiments) and 7 (Local asymptotics).NEWLINENEWLINE To be precise, in Chapters 1 and 2, the necessary matters, such as experiments admitting score and Fisher information, asymptotics of information bounds, properties of sequences of maximum likelihood, asymptotic normality of minimum distance of sequences, are explained. As a continuation, in Chapter 3, defining ``likelihood ratio'' and ``log-likelihood ratio'' for a pair \(P_n\), \(Q_n\) of probability measures, for a pair of sequences of probability measures \((P_n)\), \((Q_n)\) the relation between ``mutual contiguity'' and ``weak limit of log-likelihood'' is discussed. In Chapter 4, defining the new notion of ``smoothness'' in some sense, new and more general definitions of score and information in a statistical model are introduced and under the new setting properties of quadratic expansions of log-likelihood ratios examined. For i.i.d. models a Second LeCam Lemma is proved. In Chapter 5, defining a Gaussian shift experiment and central limit in it, it is shown that in the experiment the central statistic is a minimax estimator for the unknown parameter with respect to any subconvex loss function. Further, introducing the notion ``density process'', a Brownian motion with unknown drift is shown to be a statistical model of a Gaussian shift experiment. In the first section of Chapter 6, properties of quadratic experiments are studied and in later sections, as examples, solutions of SDE and time changes (including stopping times) for Brownian motions with unknown drift are considered. In Chapter 7, ``locally asymptotically quadratic (LAQ)'', ``locally asymptotically mixed normal (LAMN)'' and ``locally asymptotically normal (LAN)'' are defined and in the setting LAN or LAMN, the conditions, which assure estimators to be a…
description: scientific article; zbMATH DE number 6090158