Mathematics and Its Applications

Transcription

1 Stochastic Processes

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 508

3 Stochastic Processes Inference Theory by M.M. Rao University of California, Riverside, California, U.S.A. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper All Rights Reserved 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 To Professors Vlf Grenander and Tom S. Pitcher whose fundamental and deep contributions shaped stochastic inference

6 CONTENTS Preface Chapter I: Introduction and Preliminaries 1.1 The problem of inference 1.2 Testing a hypothesis 1.3 Distinguishability of hypotheses 1.4 Estimation of parameters 1.5 Inference as a decision problem 1.6 Complements and exercises Bibliographical notes Xl Chapter II: Some Principles of Hypothesis Testing Testing simple hypotheses Reduction of composite hypotheses Composite hypotheses with iterated weights Bayesian methodology for applications Further results on composite hypotheses Complements and exercises 67 Bibliographical notes 70 Chapter III: Parameter Estimation and Asymptotics Loss functions of different types Existence and other properties of estimators Some principles of estimation Asymptotics in estimation methodology Sequential estimation Complements and exercises 125 Bibliographical notes 130 vii

7 viii Chapter IV: Inferences for Classes of Processes 4.1 Testing methods for second order processes 4.2 Sequential testing of processes Weighted unbiased linear least squares prediction 4.4 Estimation in discrete parameter models 4.5 Asymptotic properties of estimators 4.6 Complements and exercises Contents Bibliographical notes 219 Chapter V: Likelihood Ratios for Processes Sets of admissible signals or translates General Gaussian processes Independent increment and jump Markov processes Infinitely divisible processes Diffusion type processes Complements and exercises 327 Bibliographical notes 335 Chapter VI: Sampling Methods for Processes 6.1 Kotel'nikov-Shannon methodology 6.2 Band limited sampling Analyticity of second order processes 6.4 Periodic sampling of processes and fields 6.5 Remarks on optional sampling 6.6 Complements and exercises Bibliographical notes Chapter VII: More on Stochastic Inference Absolute continuity of families of probability measures Likelihood ratios for families of non Gaussian measures Extension to two parameter families of measures Likelihood ratios in statistical communication theory The general Gaussian dichotomy and Girsanov's theorem Complements and exercises 452 Bibliographical notes 462 Chapter VIII: Prediction and Filtering of Processes 8.1 Predictors and projections Least squares prediction: the Cramer-Hida approach

8 Contents 8.3 Linear filtering: Bochner's formulation 8.4 Kalman-Bucy filters: the linear case 8.5 Kalman-Bucy filters: the nonlinear case 8.6 Complements and exercises Bibliographical notes ix Chapter IX: Nonparametric Estimation for Processes Spectra for classes of second order processes Asymptotically unbiased estimation of bispectra Resampling procedure and consistent estimation Associated spectral estimation for a class of processes Limit distributions of (bi)spectral function estimators Complements and exercises 593 Bibliographical notes 597 Bibliography Notation index A uthor index 633 Subject index 639

9 Preface The material accumulated and presented in this volume can be explained easily. At the start of my graduate studies in the early 1950s, I came across Grenander's (1950) thesis, and was much attracted to the entire subject considered there. I then began preparing for the necessary mathematics to appreciate and possibly make some contributions to the area. Thus after a decade of learning and some publications on the way, I wanted to write a modest monograph complementing Grenander's fundamental memoir. So I took a sabbatical leave from my teaching position at the Carnegie-Mellon University, encouraged by an Air Force Grant for the purpose, and followed by a couple of years more learning opportunity at the Institute for Advanced Study to complete the project. As I progressed, the plan grew larger needing a substantial background material which was made into an independent initial volume in (1979). In its preface I said: "My intension was to present the following material as the first part of a book treating the Inference Theory of stochastic processes, but the latter account has now receded to a distant future," namely for two more decades! Meanwhile, a much enlarged second edition of that early work has appeared (1995), and now I am able to present the main part of the original plan. In fact, while this effort took on the form of a life's project, and developing all the necessary backup material during the long gestation period, I have written some seven books and directed several theses on related topics that helped me appreciate the main subject much better. It is now termed 'stochastic inference' as an abbreviation as well as a homage to Grenander's "Stochastic processes and statistical inference". Let me explain the method adapted in preparing this work. At the outset, it became clear that there can be no compromise with the mathematics of inference theory. One observes that, broadly speaking, inference has theoretical, practical, philosophical, and interpretative aspects. But these components are also present in other scientific studies. However, for inference theory all these parts are founded on sound mathematical principles, a violation of which leads to unintended xi

10 xii Preface controversies. Thus the primary concern here is on mathematical ramifications of the subject, and the work is illustrated with a number of important examples, many of independent interest. It is noted that, as a basis of the classical statistical inference, two original sources are visible. The crucial idea on hypothesis testing is founded in the formulation of the Neyman-Pearson lemma which itself has a firm backing of the calculus of variations. All later developments of the subject are extensions of this result. Similarly, the basic idea of estimating a parameter of a distribution is founded on Fisher's formulation of the maximum likelihood (ML). The classical inference theory (for finite samples) grew out of these two fundamental principles. But the subject of stochastic processes deals with infinite collections of random variables, and there is a real barrier here to cross in order to apply the classical ideas. The necessary new accomplishment is the formulation by Grenander who showed that the (abstract) Radon Nikodym derivative must replace the likelihood function of the finite sample case, leading the way to stochastic inference. Now the determination of the general likelihood ratio (or the RN-derivative) involves an intricate analysis for which one has to employ several different technical tools. This was successfully done by Grenander himself and it was advanced further by Pitcher. Thus for a proper understanding of the subject, a greater preparation and a concerted effort are needed, and to aid this,the previous concepts are restated at various places. I shall now indicate, with some outlines of the material [more detailed summaries are at the beginning of chapters], how my original plan is executed. The first three chapters contain the basic inference theory, as described above, from the point of view of adapting it to stochastic processes. Thus the work in Chapter I, begins with the question of distinguishability (or 'identifiability') of a pair of probability measures or distributions, before a hypothesis testing question can be raised. It is better to know this fact in the theory, since, as shown by example, there exist unequal distributions which cannot be distinguished. So conditions are presented for distinguishability. Then the inference problem and its decision theoretic setup as a unifying formulation of both testing and estimation are discussed. Chapter II is devoted to detailed analysis, applications, and extensions of the Neyman-Pearson theory, and

11 xiii its many connections with other parts of analysis such as control theory and vector integral (differential) calculus, as well as extensions to composite hypotheses leading to new questions. The classical Holder, Jensen, Liapounov inequalities are shown to be consequences of this extension of the NP-Iemma. The Bayesian idea to reduce the composite hypotheses to a simple hypothesis and a simple alternative setup, as well as the hierarchical (or multistage) prior methodology are shown to lead to the hypothesis testing problems on stochastic processes. The difficulties encountered in the composite case in the original testing theory are explained vividly by a complete solution (due to Linnik) of the Behrens-Fisher problem, and this is included to show how new methods are required to answer difficult questions of inference. The detailed mathematical ideas explained in this chapter give an appreciation of inference problems leading directly to stochastic processes. Chapter III concentrates on estimation of parameters. Here optimal estimation (and prediction) relative to quadratic and more generally convex (or only increasing) loss functions is explored. Bayes estimation and (non linear) prediction are shown to be closely related. A detailed analysis including lower bounds for the corresponding risk functions is given. Further existence and asymptotic properties of ML estimators of parameters of discrete indexed processes as well as sequential sampling, introducing stopping time concepts, are studied. Considerable amount of work here appears for the first time in a book. From this point on, the rest of the chapters deal with stochastic processes. Thus the material in Chapters I-III is of general interest, and can be read even by those with less mathematical preparation, by skipping the proofs, since the general content will be appreciated by anyone interested in inference. It can also be used for a semester of graduate course. Chapters IV and V concern with major problems, of stochastic inference for classes of processes, which are based on likelihood ratios and the fundamental Neyman-Pearson-Grenander theorem. In Chapter IV, both the continuous and discrete parameter processes are treated. New techniques are needed in the continuous index case. They include Karhunen-Loeve expansions, Hellinger distances, and martingale convergence theory. These have been detailed and the work continued on second order processes with several illustrative examples. Sequential testing is considered, and the optimal character of the sequen-

12 xiv Preface tial probability ratio test for these processes is given. Here stochastic integrals and Brownian motion play key roles and they are detailed. Then weighted unbiased linear least squares prediction for stochastic flows driven by such classes as harmonizable or orthogonal increment processes are studied. In the discrete index case, sharper results are obtained for processes satisfying nth order difference equations, and properties of (least squares) estimators such as consistency and limit distributions. Chapter V deals with Gaussian and other special classes of processes. Here new tools with reproducing kernel Hilbert spaces, so perfected by Parzen, are used and Pitcher's work, on admissible means and their structure, is included. Also Gaussian dichotomy, likelihood ratios for processes of independent increments, of diffusion types, of jump Markov, and of infinitely divisible classes are studied. A number of new results, properties, and methods are presented, resulting in the longest chapter of the book. Next Chapter VI is concerned with the important problem of sampling continuous parameter second order processes. The Kotel'nikov Shannon theory and some generalizations, all for non stationary processes such as harmonizable and Cramer types, are treated. Some of the results are also extended to (isotropic) harmonizable random fields indexed by Rn, n > 1, with indications to LCA groups. Several examples are included. Chapter VII takes up extensions of some of the problems of Chapter V, for simple vs composite hypotheses and then for composite vs composite hypotheses, finding likelihood ratios in both cases. Here one has to appeal to tools such as semi-group and evolutionary operations, and the methods are somewhat abstract but quite important. These are needed for Pitcher's and his associates' works, in obtaining detailed results for such classes as seen here. An illustration to statistical communication theory is given. Another proof of Gaussian dichotomy, an extension of Girsanov's theorem, and multiple Wiener chaos are included. The latter are of interest in the recent applications of stochastic analysis to financial mathematics. Chapter VIII takes up the general concept of linear and non linear prediction, already introduced in Chapter III. In the non linear study, the existence, and strong as well as point-wise convergence of best predictors on subsets, are established when the loss function is convex employing some elementary aspects of Orlicz and Kothe space

13 xv analysis. Then for the linear least squares prediction, the Cramer-Hida approach via multiplicity theory is detailed with illustrations. Turning to linear filtering problems using Bochner's formulation, one has to obtain the unknown input observing the output when the filter is an integro-difference-differential operator and the output is, for instance, harmonizable. Characterizations are given for achieving this. Specializing the filter to signal plus noise models, sharper conditions and results can be obtained with the Kalman filtering. The main result of this theory first for discrete and then mostly for continuous parameter problems leading to stochastic differential equations (SDEs) are presented. Then the non linear theory is treated in detail. These involve just the first order SDEs, but the analysis uses classical PDEs and their stochastic counterparts, bringing this into the frontiers of research in stochastic analysis. Here, the basic (bi)spectral (or covariance) functions of the processes are needed but are often unknown to the experimenter, and they have to be estimated. So this is considered in the final Chapter IX. While the estimation of spectra are relatively well-developed for stationary processes, for the non stationary case, the main thrust here, much more work is needed. As an important illustration, solutions for a class of harmonizable processes are considered, using a resampling method, necessarily involving more observations. However, conditions for asymptotic unbiasedness, consistency, and limit distributions are obtained for bispectral density estimators. Also problems on estimation and structural analysis of processes depending on certain summability methods, isolated by Kampe de Feriet and Frenkiel, Parzen, and Rozanov are considered. Many potentially solvable problems are indicated for future research here and through out the book. Each chapter has exercises, with copious hints, complementing its work. Parts of Chapters IV-IX can be studied in graduate seminars. The numbering system is standard and is the same as in the companion volume (1995). Thus all items are serially noted, starting afresh in each section. For instance, IV.3.2 is the second item in Section 3 of Chapter IV, and in a chapter its number is dropped. In a section the chapter and section numbers are also omitted, giving only the last one. A prerequisite for studying this work is an acquaintance with real analysis, and some exposure to basic probability such as the first parts of Chapters 2, 4, and 5 of the author's text (1984). A prior knowledge of

14 xvi Preface statistics is not essential but beneficial. As already stated, this work is largely influenced by Grenander's thesis. The only published books on the topics considered here are his monograph (1981) and the two volume work by Liptser and Shriyayev (1977). However, the overlap between these and what follows is small. All sources are discussed in the Bibliographical notes. Although the presentation is reworked from publications, I have tried to credit the original authors fully, and hopefully I have been successful in this effort. In the initial stages of this project, I received helpful comments and encouragement from Tom Pitcher. A previous draft of this book, is read by Ulf Grenander and his comments, questions, suggestions, and encouragement have been invaluable. For this I express my deep gratitude to both of them. Stochastic inference uses many aspects of analysis, but also opens up many new areas. Recalling a view expressed by Hadamard in discussing Poincare's work in 1921: "the center of modern mathematics is in the theory of partial differential equations", it may be said equally that: "the center of modern probability is in the theory of stochastic inference". I hope that this point is reflected, to some extent, in the following pages. The final preparation took over two years of intense work. I do not know typing, and it is a struggle for me to compose all of this material using AMS-TeX; but somehow I did it by myself, possibly with many imperfections. Initial chapter-wise 'TeXing' was shown me by my colleague Yuichiro Kakihara. Numerous difficulties with the computer were softened by Jan Carter, and final formating and pagination were also assisted by Lambert Timmermans. My daughters Leela and Uma have given me their spare time with typing the References as well as some text. For all this help, I am very grateful. Part of the preparation of the manuscript was done with a UCR sabbatical leave last year. Finally, I hope that this book plays a role in consolidating the past and progressing into the future of inference theory. Riverside, CA January, 2000 M.M. Rao