大樣本理論基礎(chǔ)

前言

The subject of this book, first order large-sample theory, constitutes a coherent body of concepts and results that are central to both theoretical andapplied statistics. This theory underlies much of the work on such differenttopics as maximum likelihood estimation, likelihood ratio tests, the bootstrap, density estimation, contingency table analysis, and survey samplingmethodology, to mention only a few. The importance of this theory hasled to a number of books on the subject during the last 20 years, amongthem Ibragimov and Has'minskii （1979）, Serfling （1980）, Pfanzagl and Weflmeyer （1982）, Le Cam （1986）, Riischendorf （1988）, Barndorff-Nielson andCox （1989, 1994）, Le Cam and Yang （1990）, Sen and Singer （1993）, andFerguson （1996）.These books all reflect the unfortunate fact that a mathematically complete presentation of the material requires more background in probabilitythan can be expected from many students and workers in statistics. Thepresent, more elementary, volume avoids this difficulty by taking advantage of an important distinction. While the proofs of many of the theoremsrequire a substantial amount of mathematics, this is not the case with theunderstanding of the concepts and results nor of their statistical applications.

內(nèi)容概要

本書在講述一階大樣本理論方面比較獨(dú)特，討論了大量的應(yīng)用，包括密度估計(jì)、自助法和抽樣方法論的漸進(jìn)。本書的內(nèi)容比較基礎(chǔ)，適合統(tǒng)計(jì)專業(yè)的研究生和有兩年微積分背景的應(yīng)用領(lǐng)域。每章末有針對(duì)本章每節(jié)的問題和練習(xí)，每節(jié)末都附有小結(jié)。

書籍目錄

Preface 1 Mathematical Background   1.1 The concept of limit   1.2 Embedding sequences   1.3 Infinite series   1.4 Order relations and rates of convergence   1.5 Continuity   1.6 Distributions   1.7 Problems 2 Convergence in Probability and in Law   2.1 Convergence in probability   2.2 Applications   2.3 Convergence in law   2.4 The central limit theorem   2.5 Taylor's theorem and the delta method   2.6 Uniform convergence   2.7 The CLT for independent non-identical random variables   2.8 Central limit theorem for dependent variables   2.9 Problems 3 Performance of Statistical Tests   3.1 Critical values   3.2 Comparing two treatments   3.3 Power and sample size   3.4 Comparison of tests: Relative efficiency   3.5 Robustness   3.6 Problems 4 Estimation   4.1 Confidence intervals   4.2 Accuracy of point estimators   4.3 Comparing estimators   4.4 Sampling from a finite population   4.5 Problems 5 Multivariate Extensions   5.1 Convergence of multivariate distributions   5.2 The bivariate normal distribution   5.3 Some linear algebra   5.4 The multivariate normal distribution   5.5 Some applications   5.6 Estimation and testing in 2 × 2 tables   5.7 Testing goodness of fit   5.8 Problems 6 Nonparametric Estimation   6.1 U-Statistics   6.2 Statistical functionals   6.3 Limit distributions of statistical functionals   6.4 Density estimation   6.5 Bootstrapping   6.6 Problems 7 Efficient Estimators and Tests   7.1 Maximum likelihood   7.2 Fisher information   7.3 Asymptotic normality and multiple roots   7.4 Efficiency   7.5 The multiparameter case I. Asymptotic normality   7.6 The multiparameter case II. Efficiency   7.7 Tests and confidence intervals   7.8 Contingency tables   7.9 Problems Appendix References Author Index Subject Index

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