Asymptotics in statistics. Some basic concepts.
2nd ed.

*(English)*Zbl 0952.62002
Springer Series in Statistics. New York, NY: Springer. xiii, 285 p. DM 139.00; öS 1015.00; sFr 126.50; £48.00; $ 69.95 (2000).

The 1990 edition, see the review Zbl 0719.62003, was written based on L. Le Cam lectures at the University of Montreal in the summer of 1968 [Théorie asymptotique de la décision statistique. (1969; Zbl 0374.62024)]. This is a second edition “reviewed and enlarged” by the authors, aiming at being understood about LAN and related topics by a “good” second-year graduate student.

Let \({\mathcal E}= \{\theta\in\Theta\}\) be a family of probability measures \(P_\theta\) carried by a \(\sigma\)-field \({\mathcal A}\) of a set \({\mathcal X}\). We call \({\mathcal E}\) the experiment. In this book many asymptotic problems in statistics are considered by the experiments. In Chapters 2 and 3, deficiency and distance between experiments are introduced. In Chapter 4, Gaussian and Poisson experiments are defined and some properties are considered. In Chapter 5, limit laws for likelihood ratios, such as limits for binary experiments and Gaussian limits are treated. In Chapters 6 and 7, the notions of LAN and LAMN are introduced and their properties and applications are studied. In Chapter 8, some asymptotic properties of Bayes procedures are discussed. This book does not contain topics of empirical processes.

Let \({\mathcal E}= \{\theta\in\Theta\}\) be a family of probability measures \(P_\theta\) carried by a \(\sigma\)-field \({\mathcal A}\) of a set \({\mathcal X}\). We call \({\mathcal E}\) the experiment. In this book many asymptotic problems in statistics are considered by the experiments. In Chapters 2 and 3, deficiency and distance between experiments are introduced. In Chapter 4, Gaussian and Poisson experiments are defined and some properties are considered. In Chapter 5, limit laws for likelihood ratios, such as limits for binary experiments and Gaussian limits are treated. In Chapters 6 and 7, the notions of LAN and LAMN are introduced and their properties and applications are studied. In Chapter 8, some asymptotic properties of Bayes procedures are discussed. This book does not contain topics of empirical processes.

Reviewer: K.-i.Yoshihara (Tokyo)