Large Sample Techniques for Statistics
This book offers a comprehensive guide to large sample techniques in statistics. More importantly, it focuses on thinking skills rather than just what formulae to use; it provides motivations, and intuition, rather than detailed proofs; it begins with ver
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Jiming Jiang
Large Sample Techniques for Statistics
1C
Jiming Jiang University of California Department of Statistics 1 Shields Avenue Davis, California 95616 USA [email protected] STS Editorial Board George Casella Department of Statistics University of Florida Gainesville, FL 32611-8545 USA
Stephen Fienberg Department of Statistics Carnegie Mellon University Pittsburg, PA 15213-3890 USA
Ingram Olkin Department of Statistics Stanford University Stanford, CA 94305 USA
ISSN 1431-875X ISBN 978-1-4419-6826-5 e-ISBN 978-1-4419-6827-2 DOI 10.1007/978-1-4419-6827-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010930134 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
For my parents, Huifen and Haoliang, and my sisters, Qiuming and Dongming, with love
Preface
In a way, the world is made up of approximations, and surely there is no exception in the world of statistics. In fact, approximations, especially large sample approximations, are very important parts of both theoretical and applied statistics. The Gaussian distribution, also known as the normal distribution, is merely one such example, due to the well-known central limit theorem. Large-sample techniques provide solutions to many practical problems; they simplify our solutions to difficult, sometimes intractable problems; they justify our solutions; and they guide us to directions of improvements. On the other hand, just because large-sample approximations are used everywhere, and every day, it does not guarantee that they are used properly, and, when the techniques are misused, there may be serious consequences. Example 1 (Asymptotic χ2 distribution). Likelihood ratio test (LRT) is one of the fundamental techniques in statistics. It is well known that, in the “standard” situation, the asymptotic null distribution of the LRT is χ2 , with the degrees of freedom equal to the difference between the dimensions, defined as the numbers of free parameters, of the two nested models being compared (e.g., Rice 1995, pp. 310). This might lead to a wrong impression that the asymptotic (null) distribution of the LRT is always χ2 . A simila
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