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2000 | Book | 2. edition

Introduction Read chapter Read first chapter

Asymptotics in Statistics

Some Basic Concepts

Authors: Lucien Le Cam, Grace Lo Yang

Publisher: Springer New York

Book Series : Springer Series in Statistics

Included in: Professional Book Archive

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About this book

by Sara van de Geer. Also, we did not include material due to David Donoho, lain Johnstone, and their school. We found our­ selves unprepared to write a distillate of the material. We did touch briefly on "nonparametrics," but not on "semiparamet­ rics." This is because we feel that the semiparametric situation has not yet been properly structured. We hope that the reader will find this book interesting and challenging, in spite of its shortcomings. The material was typed in LaTeX form by the authors them­ selves, borrowing liberally from the 1990 script by Chris Bush. It was reviewed anonymously by distinguished colleagues. We thank them for their kind encouragement. Very special thanks are due to Professor David Pollard who took time out of a busy schedule to give us a long list of suggestions. We did not follow them all, but we at least made attempts. We wish also to thank the staff of Springer-Verlag for their help, in particular editor John Kimmel, who tried to make us work with all deliberate speed. Thanks are due to Paul Smith, Te-Ching Chen and Ju-Yi-Yen, who helped with the last-minute editorial corrections.

Table of Contents

Frontmatter
1. Introduction
Abstract
In this volume we describe a few concepts and tools that we have found useful in thinking about asymptotic problems in statistics.
Lucien Le Cam, Grace Lo Yang
2. Experiments, Deficiencies, Distances
Abstract
Following Blackwell [1951] we shall call experiment a family ε = { Pθ; θ ∈ Θ } of probability measures P δon a σ-field A of subsets of a set χ. The set of indices is called the parameter space. To obtain a statistical decision problem in the sense of Wald we also need a set Z of possible decisions and a loss function W defined on Θ × Z and taking values in (-∞, +∞].
Lucien Le Cam, Grace Lo Yang
3. Contiguity — Hellinger Transforms
Abstract
We saw, in Chapter 2, that when Θ is finite, convergence of experiments in the sense of our distance is equivalent to convergence in distribution of likelihood ratios. Here we shall describe, in Section 3.1, some consequences of a condition, called contiguity that simplifies many arguments in passages to the limit. Contiguity is simply an asymptotic form of absolute continuity. Theorem 1 establishes several equivalent forms of the conditions for contiguity. One of the most useful consequences of contiguity is its application to the joint limiting distribution of statistics and likelihood ratios, described in Proposition 1. We also introduce, in Section 3.2, a technical tool, the Hellinger transform, that is often convenient in studies involving independent observations. In that case it provides the same flexibility as that given by characteristic functions in the study of sums of independent random variables.
Lucien Le Cam, Grace Lo Yang
4. Gaussian Shift and Poisson Experiments
Abstract
This chapter is an introduction to some experiments that occur in their own right but are also very often encountered as limits of other experiments. The gaussian ones have been used and overused because of their mathematical tractability, not just because of the Central Limit Theorem. The Poisson experiments cannot be avoided when one models natural phenomena. We start with the common gaussian experiments. By “gaussian” we shall understand throughout they are the “gaussian shift” experiments, also called homoschedastic. There is a historical reason for that appellation: In 1809 Gauss introduced them (in the one-dimensional case) as those experiments where the maximum likelihood estimate coincides with the mean of the observations. This fact, however, will not concern us.
Lucien Le Cam, Grace Lo Yang
5. Limit Laws for Likelihood Ratios
Abstract
In this chapter we shall consider a double sequence { ε n,j ; j = 1, 2, …; n = 1, 2, … } of experiments ε n,j = { p t,n,j ; t ∈ Θ }. Let be ε n the direct product in j of the hat is, n consists of performing the independently of each other. The measures that constitute ε n are the product measures Pt,n = IIj; Pt,n,j • It will usually be assumed that j runs through a finite set.
Lucien Le Cam, Grace Yang
6. Local Asymptotic Normality
Abstract
The classical theory of asymptotics in statistics relies heavily on certain local quadratic approximations to the logarithms of likelihood ratios. Such approximations will be studied here but in a restricted framework.
Lucien Le Cam, Grace Lo Yang
7. Independent, Identically Distributed Observations
Abstract
The structure that has received by far the most attention in the statistical literature is one that can be described as follows. One takes a family of probability measures { pθ: θ ∈ Θ } on some space (X, A). Then one considers experiments ε n = { Pθ, n; θ ∈ Θ } where Pθ, n is the joint distribution of n observations X1, X2, …, X n all independent with individual distribution p θ. One studies the asymptotic behavior of the system as n tends to infinity. We shall refer to this as the “standard i.i.d. case.”
Lucien Le Cam, Grace Lo Yang
8. On Bayes Procedures
Abstract
In this chapter we describe some of the asymptotic properties of Bayes procedures. These are obtained by using on the parameter set Θ a finite positive measure p and minimizing the average risk ∫ R (θ, ρ) μ (dθ). (See Chapter 2 for notation.) The procedure p that achieves this minimum will, of course, depend on the choice of μ. However, the literature contains numerous statements to the effect that, for large samples, the choice of μ matters little. This cannot be generally true, but we start with a proposition to this effect. If instead of μ one uses λ dominated by μ and if the density dλ/dp can be closely estimated, then a procedure that is nearly Bayes for p is also nearly Bayes for λ.
Lucien Le Cam, Grace Lo Yang
Backmatter
Metadata
Title
Asymptotics in Statistics
Authors
Lucien Le Cam
Grace Lo Yang
Copyright Year
2000
Publisher
Springer New York
Electronic ISBN
978-1-4612-1166-2
Print ISBN
978-1-4612-7030-0
DOI
https://doi.org/10.1007/978-1-4612-1166-2