دانلود کتاب Mathematical Theory of Statistics

The present book is neither intended to be a text-book for a course in statistics, nor a monograph on statistical decision theory. It is rather the attempt to connect well-known facts of classical statistics with problems and methods of contemporary research. To justify such a connection, let us recapitulate some modern developments of mathematical statistics. A considerable amount of statistical research during the last thirty years can be subsumed under the label "asymptotic justification of statistical methods". Already in the first half of this century asymptotic arguments became neces- sary, as it turned out that the optimization problems of statistics can he solved for finite sample size only in very few particular cases. The starting point of a systematic mathematical approach to asymptotics was the famous paper by Wald [1943]. This early paper contains already many important ideas for the asymptotic treatment of testing hypotheses. Ten years later the thesis of LeCam appeared, which in the meantime has turned out to be leading for asymptotic estimation. Briefly, the situation considered in both papers is as follows. As mathematical model serves a family of probability measures or, for short, an experiment which can be viewed as a finite dimensional differentiable manifold, i.e. a so-called parametric model. By independent replication of the experiment it becomes possible to restrict the analysis to a small subset of the model. i,e. to localize the problem. As the sample size tends to infinity the experiment can locally be approximated by a much simpler one, namely by a Gaussian shift. Essentially, this is due to the smoothness properties of the experiment. Thus, for large sample sizes the statistical analysis of the original experiment can be replaced by the analysis of the approximating Gaussian shift. It took at least twenty years to understand the structure of the papers of Wald and LeCam in the way described above. Remarkable steps were the papers by LeCam [1960], and by Hajek [1970 and 1972]. Finally, LeCam succeeded in extending his version of decision theory [1955 and 1964] to an asymptotic theory of experiments [1972 and 1979]. It covers the main results of asymptotic statistics obtained so far, thus providing a framework which facilitates the understanding of classical asymptotics considerably. Moreover, for good reasons this theory claims to determine the framework of future developments in asymptotic statistics. Apart from the above-mentioned papers the asymptotic theory of experi- ments is presented in Lecture Notes of LeCam, [1969 and 1974] and hopefully soon by a forthcoming monograph of LeCam. The present book is intended to serve as a "missing ]ink" between introductory text-books like those of Lehmann [1958], Schmetterer 11974], or Witting L1985], and the presentation by LeCam. This goal determines what the book contains as well as the omissions. As to the mathematical prerequisites, we present the asymptotic theory of experiments on a mathematicalleve] corresponding to the usual level of upper graduate courses in probability. Essentially, there are two sets of problems where the state of the art justifies the attempt of unified presentation. First, there is the genera] decision-theoretic framework of statistics together with asymptotic decision theory, and second, its application to the case of independent replications of parametric models. The present volume deals mainly with the first complex. To give a rough outline of the organization of the book, let us discuss briefly the contents. More detailed information is provided by the introductions to the single chapters. After having collected some more or less well-known facl of probability theory in Chapter 1, we deal in Chapter 2 with basic facts of testing hypotheses. On the one hand, the results of this chapter are needed later, but on the other hand they convey a first idea of handling decision-theoretic concepts. It turns out, that the theory of Neyman-Pearson is the basic tool for the analysis of binary experiments. This gives rise to exemplify the theory of experiments in Chapter 3 by means of these tools only. Solely by the theory of Neyman and Pearson some of the main results of the theory of experiments are proved in an elementary way. Already classical statistics compares experiments with respect to the informa- tion contained. The central role plays the concept of sufficiency. In Chapter 4 we take up this idea and extend it, passing the notion of exhaustivity, to the general concept of randomization of dominated experiments. By relatively simple methods we prove a version of the randomization criterion for domi- nated experiments. In Chapter 5 we coHect the most important app1ications of sufficiency to exponential experiments and Gaussian shifts. The testing problems which have been considered up to this point are of dimension one. In Chapter 6 we start with the consideration of higher dimensional testing problems. We begin with Wald's classical complete class theorem and prove the completeness of convex acceptance regions for ex- ponential experiments. Our main interest in this chapter are Gaussian shift experiments of dimension greater than one. At first, some basic facts are proved elementary, i.e. by means of the Neyman-Pearson lemma. The rest of the chapter, however, is devoted to another approach to testing for Gaussian shifts, namely to reduction by invariance. In Chapter 6 we take this opportunity for discussing amenability of groups and for proving the most simple version of the Hunt-Stein theorem. Partly, these results are needed later, but their presenta- tion in Chapter 6 serves mainly as an introduction to the ideas of statistical invariance concepts. Before we go into the general decision theory we present in Chapter 7 a brief compendium of estimation theory. The power of the Neyman-Pearson theory is illustrated by median unbiased estimates. Sufficiency is applied to mean unbiased estimates. As it becomes clear already in Chapter 3, the natural extension of the Neyman-Pearson lemma to arbitrary experiments leads to the Bayesian calculus which is considered without any regard to "subjectivistic" interpretation. It turns out that invariance-theoretic methods of estimation as well as most proofs of admissibility are, in a technical sense, Bayesian in spirit. At this point there is plenty of motivation, to introduce the concepts of decision theory in full generality. In Chapter 8 we present the classical results of decision theory, such as the minimax theorem, the complete class theorem and the general Hunt-Stein theorem. Chapter 9 deals with those parts of decision theory which are known under the label of comparison of experiments. The general theorems of asymptotic decision theory are contained in Chapter 10. By means of these results it is possible to carry through our program. namely to reduce asymptotic problems to the analysis of the limit experiments. This is done in the remaining Chapter 11-13. Let us have a closer look at the program. The asymptotic method consists of three steps corresponding to Chapters 11. 12 and 13. The main idea is to embed a statistical experiment into a convergent sequence of experiments and then to analyse the limit experiment of the sequence instead of the original one. Hence, one major problem is the statistical analysis of the limit experiment. In the present book we confine ourselves to limits which arc Gaussian shifts. In Chapter 11 the statistical theory of Gaussian shifts is recapitulated sufficiently general to cover infinite dimensional parameter spaces, A second problem consists in the proof of convergence for certain sequences of experiments. In this context we only consider the case of convergence to Gaussian shifts. This case can be treated by means of stochastic expansions of likelihood processes. By way of example we establish in Chapter 12 the basic expansion for independent replications of a given experiment. As soon as convergence of a sequence against a limit experiment is established the asymptotic decision theory of Chapter 10 may be applied. This synthesis is carried through in Chapter 13 for sequences of experiments converging to Gaussian shifts. In this way, we obtain the main results of classical asymptotics. We show at hand of examples how to treat both parametric as well as non parametric problems by the tools established so far. lL is clear that the motivation for the development of a general asymptotic decision theory can only be found in classical examples, which are not the main subject of this book. The connoisseur of classical statistics will not mind this lack of discussing the intuitive background of the concepts. But for the beginner we emphasize that this book will completely miss its aim if the reader is not aware of the connection with the origin of the ideas. There is a sufficient number of good textbooks covering c1assical statistical methods and some asymptotics, which should be used as a pennanent reference. Moreover, it is highly recommended to read the original papers quoted in the text. We do not claim that our presentation is final in any respect but we hope that it will be helpful for some reader.