Asymptotic statistics. With a view to stochastic processes
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 asymptotically optimal, are considered. In Chapter 8, some stochastic processes in which arise LAN or LAMN or LAQ, such as Ornstein-Uhlenbeck processes with unknown parameter, null recurrent diffusion models, etc., are discussed.NEWLINENEWLINE Many examples and exercises are given. This book is good for graduate students and researchers interested in statistics of stochatic processes.
scientific article; zbMATH DE number 6090158