Theoretical statistics

"A text that stresses the general concepts of the theory of statistics Theoretical Statistics provides a systematic statement of the theory of statistics, emphasizing general concepts rather than mathematical rigor. Chapters 1 through 3 provide an overview of statistics and discuss some of the basic philosophical ideas and problems behind statistical procedures. Chapters 4 and 5 cover hypothesis testing with simple and null hypotheses, respectively. Subsequent chapters discuss non-parametrics, interval estimation, point estimation, asymptotics, Bayesian procedure, and deviation theory. Student familiarity with standard statistical techniques is assumed."--Provided by publisher Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Dedication 6 Table of Contents 8 Preface 12 Chapter 1: Introduction 14 1.1 Objectives of statistical analysis and theory 14 1.2 Criteria for the choice of families of models 18 1.3 The analysis of complex responses 20 1.4 Plan of the book 21 Bibliographic notes 23 Chapter 2: Some general concepts 24 2.1 The likelihood 24 2.2 Sufficient statistics 31 2.3 Some general principles of statistical inference 49 2.4 Some approaches to statistical inference 58 Bibliographic notes 69 Further results and exercises 71 Chapter 3: Pure significance tests 77 3.1 Null hypotheses 77 3.2 Test statistics and their null distributions 79 3.3 Composite null hypotheses 86 3.4 Discussion 89 Bibliographic notes 95 Further results and exercises 96 Chapter 4: Significance tests: simple null hypotheses 101 4.1 General 101 4.2 Formulation 103 4.3 Simple null hypothesis and simple alternative hypothesis 104 4.4 Some examples 106 4.5 Discrete problems 112 4.6 Composite alternatives 114 4.7 Two-sided tests 118 4.8 Local power 119 4.9 Multidimensional alternatives 134 Bibliographic notes 138 Further results and exercises 139 Chapter 5: Significance tests: composite null hypotheses 144 5.1 General 144 5.2 Similar regions 147 5.3 Invariant tests 170 5.4 Some more difficult problems 184 Bibliographic notes 187 Further results and exercises 188 Chapter 6: Distribution-free and randomization tests 192 6.1 General 192 6.2 Permutation tests 195 6.3 Rank tests 200 6.4 Randomization tests 209 6.5 Distance tests 211 Bibliographic notes 215 Further results and exercises 216 Chapter 7: Interval estimation 220 7.1 Introduction 220 7.2 Scalar parameter 221 7.3 Scalar parameter with nuisance parameters 241 7.4 Vector parameter 249 7.5 Estimation of future observations 255 Bibliographic notes 259 Further results and exercises 260 Chapter 8: Point estimation 263 8.1 General 263 8.2 General considerations on bias and variance 265 8.3 Cramér-Rao inequality 267 8.4 Achievement of minimum variance and removal of bias 271 8.5 Estimates of minimum mean squared error 279 8.6 Robust estimation 283 Bibliographic notes 285 Further results and exercises 286 Chapter 9: Asymptotic theory 292 9.1 Introduction 292 9.2 Maximum likelihood estimates 296 9.3 Large-sample parametric significance tests 324 9.4 Robust inference for location parameters 357 Bibliographic notes 367 Further results and exercises 369 Chapter 10: Bayesian methods 377 10.1 Introduction 377 10.2 Bayes’s theorem 378 10.3 Conjugate prior distributions 382 10.4 Interpretation of Bayesian probability statements 388 10.5 Bayesian versions of some previous types of procedure 403 10.6 Asymptotic Bayesian theory 412 10.7 Empirical Bayes procedures 413 Bibliographic notes 419 Further results and exercises 419 Chapter 11: Decision theory 425 11.1 Examples of statistical decision problems 425 11.2 Formulation of terminal decision problems 428 11.3 Solution of the fully specified terminal decision problem 430 11.4 Utility 435 11.5 Incompletely specified decision problems 439 11.6 Decision theory without prior distributions 442 11.7 Decision theory representation of common problems 454 11.8 A remarkable case of inadmissibility 458 11.9 Sequential decision problems 464 Bibliographic notes 471 Further results and exercises 472 Appendix 1: Determination of probability distributions 475 Appendix 2: Order statistics 479 A2A General properties 479 A2.2 Special distributions 480 A2.3 Asymptotic distributions 481 A2.4 Linear combinations of order statistics 483 A2.5 Extreme value theory 485 Bibliographic notes 487 Appendix 3: Second-order regression for arbitrary random variables 488 References 491 Author Index 509 Subject Index 512 This text provides a systematic statement of the theory of statistics, with emphasis placed on general concepts rather than on mathematical rigour. A familiarity with standard statistical techniques is assumed.