Selected Works of Peter J. Bickel
This volume presents selections of Peter J. Bickel’s major papers, along with comments on their novelty and impact on the subsequent development of statistics as a discipline. Each of the eight parts concerns a particular area of research and provides new
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Asymptotic Theory Qi-Man Shao
3.1 Introduction to Four Papers on Asymptotic Theory 3.1.1 General Introduction Asymptotic theory plays a fundamental role in the developments of modern statistics, especially in the theoretical analysis of new methodologies. Some asymptotic results may borrow directly from the limit theory in probability, but many need deep insights of statistical contents and more accurate approximations, which have in turn fostered further developments of limit theory in probability. Peter Bickel has made far-reaching and wide-ranging contributions to modern statistics. He is a giant in theoretical statistics. In asymptotic theory, besides his contributions to bootstrap and high-dimensional statistical inference, in this paper I shall focus on four of his seminal papers on asymptotic expansions and Bartlett correction for Bayes solutions, likelihood ratio statistics and maximum-likelihood estimator for general hidden Markov models. The papers will be reviewed in chronological order.
3.1.2 Asymptotic Theory of Bayes Solutions The paper of Bickel and Yahav (1969) deals with the asymptotic theory of Bayes solutions in estimation and hypothesis testing. It proves that Bayes estimates arising from a loss function are asymptotically efficient and that the mean of the posterior distribution is asymptotically normal, which confirms a long time statistical folklore.
Q.-M. Shao () Department of Statistics, Chinese University of Hong Kong, Shatin, Kowloon, Hong Kong e-mail: [email protected] J. Fan et al. (eds.), Selected Works of Peter J. Bickel, Selected Works in Probability and Statistics 13, DOI 10.1007/978-1-4614-5544-8 3, © Springer Science+Business Media New York 2013
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The results also significantly extend some early work of Le Cam. More importantly, the paper provides asymptotic expansions for the posterior risk in the estimation problem. The expansion can be viewed in the same sprit of Badahur’s work, which is now commonly called the Bahadur representation. It is noted that Bickel and Yahav derived the expansion from an entirely different viewpoint. The method, setup and results in Bickel and Yahav (1969) have a significant impact to the later wok in this area directly and indirectly. For example, Yuan (2009) proposed a joint estimation procedure in which some of the parameters are estimated Bayesian, and the rest by the maximum-likelihood estimator in the same parametric model. The proof of the consistency of the hybrid estimate is based on the method in Bickel and Yahav (1969). The paper of He and Shao (1996) on the Bahadur expansion for M-estimators follows a similar setup as Bickel and Yahav (1969). Belloni and Chernozhukov (2009) also follow the setup of B-Y and extend some of their results. The results of Bickel and Yahav (1969) have considerable applications in asymptotic sequential analysis. For recent results and extensions on this topic we refer to Hwang (1997), Ghosal (1999), and Belloni and Chernozhukov (2009) and references therein.
3.1.3 The Bartlett Correction T
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