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Preface page xi
Acknowledgments xxiv
1 An introduction to empirical modeling 1
1.1 Introduction 1
1.2 Stochastic phenomena, a preliminary view 3
1.3 Chance regularity and statistical models 13
1.4 Statistical adequacy 16
1.5 Statistical versus theory information* 19
1.6 Observed data 20
1.7 Looking ahead 29
1.8 Exercises 30
2 Probability theory: a modeling framework 31
2.1 Introduction 31
2.2 Simple statistical model: a preliminary view 33
2.3 Probability theory: an introduction 39
2.4 Random experiments 42
2.5 Formalizing condition [a]: the outcomes set 45
2.6 Formalizing condition [b]: events and probabilities 48
2.7 Formalizing condition [c]: random trials 69
2.8 Statistical space 73
2.9 A look forward 74
2.10 Exercises 75
3 The notion of a probability model 77
3.1 Introduction 77
3.2 The notion of a simple random variable 78
3.3 The general notion of a random variable 85
3.4 The cumulative distribution and density functions 89
v
3.5 From a probability space to a probability model 97
3.6 Parameters and moments 104
3.7 Moments 109
3.8 Inequalities 131
3.9 Summary 132
3.10 Exercises 133
Appendix A Univariate probability models 135
A.1 Discrete univariate distributions 136
A.2 Continuous univariate distributions 138
4 The notion of a random sample 145
4.1 Introduction 145
4.2 Joint distributions 147
4.3 Marginal distributions 155
4.4 Conditional distributions 158
4.5 Independence 167
4.6 Identical distributions 171
4.7 A simple statistical model in empirical modeling: a preliminary view 175
4.8 Ordered random samples* 181
4.9 Summary 184
4.10 Exercises 184
Appendix B Bivariate distributions 185
B.1 Discrete bivariate distributions 185
B.2 Continuous bivariate distributions 186
5 Probabilistic concepts and real data 190
5.1 Introduction 190
5.2 Early developments 193
5.3 Graphic displays: a t-plot 195
5.4 Assessing distribution assumptions 197
5.5 Independence and the t-plot 212
5.6 Homogeneity and the t-plot 217
5.7 The empirical cdf and related graphs* 229
5.8 Generating pseudo-random numbers* 254
5.9 Summary 258
5.10 Exercises 259
6 The notion of a non-random sample 260
6.1 Introduction 260
6.2 Non-random sample: a preliminary view 263
6.3 Dependence between two random variables: joint distributions 269
6.4 Dependence between two random variables: moments 272
6.5 Dependence and the measurement system 282
6.6 Joint distributions and dependence 290
vi Contents
6.7 From probabilistic concepts to observed data 309
6.8 What comes next? 330
6.9 Exercises 335
7 Regression and related notions 337
7.1 Introduction 337
7.2 Conditioning and regression 339
7.3 Reduction and stochastic conditioning 356
7.4 Weak exogeneity* 366
7.5 The notion of a statistical generating mechanism (GM) 368
7.6 The biometric tradition in statistics 377
7.7 Summary 397
7.8 Exercises 397
8 Stochastic processes 400
8.1 Introduction 400
8.2 The notion of a stochastic process 403
8.3 Stochastic processes: a preliminary view 410
8.4 Dependence restrictions 420
8.5 Homogeneity restrictions 426
8.6 “Building block” stochastic processes 431
8.7 Markov processes 433
8.8 Random walk processes 435
8.9 Martingale processes 438
8.10 Gaussian processes 444
8.11 Point processes 458
8.12 Exercises 460
9 Limit theorems 462
9.1 Introduction to limit theorems 462
9.2 Tracing the roots of limit theorems 465
9.3 The Weak Law of Large Numbers 469
9.4 The Strong Law of Large Numbers 476
9.5 The Law of Iterated Logarithm* 481
9.6 The Central Limit Theorem 482
9.7 Extending the limit theorems* 491
9.8 Functional Central Limit Theorem* 495
9.9 Modes of convergence 503
9.10 Summary and conclusion 510
9.11 Exercises 510
10 From probability theory to statistical inference* 512
10.1 Introduction 512
10.2 Interpretations of probability 514
Contents vii
10.3 Attempts to build a bridge between probability and observed data 520
10.4 Toward a tentative bridge 528
10.5 The probabilistic reduction approach to specification 541
10.6 Parametric versus non-parametric models 546
10.7 Summary and conclusions 556
10.8 Exercises 556
11 An introduction to statistical inference 558
11.1 Introduction 558
11.2 An introduction to the classical approach 559
11.3 The classical versus the Bayesian approach 568
11.4 Experimental versus observational data 570
11.5 Neglected facets of statistical inference 575
11.6 Sampling distributions 578
11.7 Functions of random variables 584
11.8 Computer intensive techniques for approximating sampling
distributions* 594
11.9 Exercises 600
12 Estimation I: Properties of estimators 602
12.1 Introduction 602
12.2 Defining an estimator 603
12.3 Finite sample properties 607
12.4 Asymptotic properties 615
12.5 The simple Normal model 621
12.6 Sufficient statistics and optimal estimators* 627
12.7 What comes next? 635
12.8 Exercises 635
13 Estimation II: Methods of estimation 637
13.1 Introduction 637
13.2 Moment matching principle 639
13.3 The least-squares method 648
13.4 The method of moments 654
13.5 The maximum likelihood method 659
13.6 Exercises 678
14 Hypothesis testing 681
14.1 Introduction 681
14.2 Leading up to the Fisher approach 682
14.3 The Neyman–Pearson framework 692
14.4 Asymptotic test procedures* 713
14.5 Fisher versus Neyman–Pearson 720
viii Contents
14.6 Conclusion 727
14.7 Exercises 727
15 Misspecification testing 729
15.1 Introduction 729
15.2 Misspecification testing: formulating the problem 733
15.3 A smorgasbord of misspecification tests 739
15.4 The probabilistic reduction approach and misspecification 753
15.5 Empirical examples 765
15.6 Conclusion 783
15.7 Exercises 784
References 787
Index 806
Contents ix
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