Theory of Statistical Inference

470 Pages·2021·11.422 MB·English

by  Anthony Almudevar

Theory of Statistical
Inference
CHAPMAN & HALL/CRC
Texts in Statistical Science Series
Joseph K. Blitzstein, Harvard University, USA
Julian J. Faraway, University of Bath, UK
Martin Tanner, Northwestern University, USA
Jim Zidek, University of British Columbia, Canada
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Theory of Statistical
Inference
Anthony Almudevar
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Library of Congress Cataloging-in-Publication Data
Names: Almudevar, Anthony, author.
Title: Theory of statistical inference / Anthony Almudevar.
Description: Boca Raton : CRC Press, 2021. | Series: Chapman & Hall/CRC texts
in statistical science | Includes bibliographical references and index.
Identifiers: LCCN 2021032159 (print) | LCCN 2021032160 (ebook) | ISBN
9780367488758 (hardback) | ISBN 9780367502805 (paperback) | ISBN
9781003049340 (ebook)
Subjects: LCSH: Mathematical statistics.
Classification: LCC QA276 .A46 2021 (print) | LCC QA276 (ebook) | DDC
519.5/4--dc23
LC record available at https://lccn.loc.gov/2021032159
LC ebook record available at https://lccn.loc.gov/2021032160
ISBN: 9780367488758 (hbk)
ISBN: 9780367502805 (pbk)
ISBN: 9781003049340 (ebk)
DOI: 10.1201/9781003049340
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Contents
Preface xi
1 Distribution Theory 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Some Important Theorems of Probability . . . . . . . . . . . . . . . . . . 7
1.4 Commonly Used Distributions . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Stochastic Order Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Inversion of the CDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.8 Transformations of Random Variables . . . . . . . . . . . . . . . . . . . . 22
1.9 Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . 23
1.10 Moments and Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Multivariate Distributions 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Parametric Classes of Multivariate Distributions . . . . . . . . . . . . . . 37
2.3 Multivariate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 Quadratic Forms, Idempotent Matrices and Cochran’s Theorem . . . . . . 45
2.6 MGF and CGF of Independent Sums . . . . . . . . . . . . . . . . . . . . . 49
2.7 Multivariate Extensions of the MGF . . . . . . . . . . . . . . . . . . . . . 51
2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Statistical Models 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Parametric Families for Statistical Inference . . . . . . . . . . . . . . . . . 58
3.3 Location-Scale Parameter Models . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Regular Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Fisher Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 Exponential Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8 Complete and Ancillary Statistics . . . . . . . . . . . . . . . . . . . . . . . 82
3.9 Conditional Models and Contingency Tables . . . . . . . . . . . . . . . . . 88
3.10 Bayesian Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.11 Indifference, Invariance and Bayesian Prior Distributions . . . . . . . . . . 91
3.12 Nuisance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.13 Principles of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
v
vi Contents
4 Methods of Estimation 105
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Unbiased Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3 Method of Moments Estimators . . . . . . . . . . . . . . . . . . . . . . . . 107
4.4 Sample Quantiles and Percentiles . . . . . . . . . . . . . . . . . . . . . . . 108
4.5 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . 109
4.6 Confidence Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.7 Equivariant Versus Shrinkage Estimation . . . . . . . . . . . . . . . . . . 123
4.8 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 Hypothesis Testing 133
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Principles of Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4 The Observed Level of Significance (P-Values) . . . . . . . . . . . . . . . 137
5.5 One- and Two-Sided Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.6 Unbiasedness and Stochastic Ordering . . . . . . . . . . . . . . . . . . . . 139
5.7 Hypothesis Tests and Pivots . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.8 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.9 Similar Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6 Linear Models 155
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.2 Linear Models – Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 Best Linear Unbiased Estimators (BLUE) . . . . . . . . . . . . . . . . . . 158
6.4 Least Squares Estimators, BLUEs and Projection Matrices . . . . . . . . 161
6.5 Ordinary and Generalized Least Squares Estimators . . . . . . . . . . . . 163
6.6 ANOVA Decomposition and the F Test for Linear Models . . . . . . . . . 168
6.7 One- and Two-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.8 Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.9 Constrained Least Squares Estimation . . . . . . . . . . . . . . . . . . . . 187
6.10 Simultaneous Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . 191
6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7 Decision Theory 207
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2 Ranking Estimators by MSE . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.4 The Structure of Decision Theoretic Inference . . . . . . . . . . . . . . . . 215
7.5 Loss and Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.6 Uniformly Minimum Risk Estimators (The Location-Scale Model) . . . . 221
7.7 Some Principles of Admissibility . . . . . . . . . . . . . . . . . . . . . . . 224
7.8 Admissibility for Exponential Families (Karlin’s Theorem) . . . . . . . . . 226
7.9 Bayes Decision Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
7.10 Admissibility and Optimality . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Contents vii
8 Uniformly Minimum Variance Unbiased (UMVU) Estimation 241
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8.2 Definition of UMVUE’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.3 UMVUE’s and Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.4 Methods of Deriving UMVUEs . . . . . . . . . . . . . . . . . . . . . . . . 245
8.5 Nonparametric Estimation and U-statistics . . . . . . . . . . . . . . . . . 247
8.6 Rank Based Measures of Correlation . . . . . . . . . . . . . . . . . . . . . 252
8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
9 Group Structure and Invariant Inference 257
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.2 MRE Estimators for Location Parameters . . . . . . . . . . . . . . . . . . 258
9.3 MRE Estimators for Scale Parameters . . . . . . . . . . . . . . . . . . . . 265
9.4 Invariant Density Families . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
9.5 Some Applications of Invariance . . . . . . . . . . . . . . . . . . . . . . . 274
9.6 Invariant Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
10 The Neyman-Pearson Lemma 289
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.2 Hypothesis Tests as Decision Rules . . . . . . . . . . . . . . . . . . . . . . 289
10.3 Neyman-Pearson (NP) Tests . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.4 Monotone Likelihood Ratios (MLR) . . . . . . . . . . . . . . . . . . . . . 294
10.5 The Generalized Neyman-Pearson Lemma . . . . . . . . . . . . . . . . . . 296
10.6 Invariant Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
10.7 Permutation Invariant Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 304
10.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
11 Limit Theorems 315
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
11.2 Limits of Sequences of Random Variables . . . . . . . . . . . . . . . . . . 316
11.3 Limits of Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
11.4 Uniform Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
11.5 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 321
11.6 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
11.7 Multivariate Extensions of Limit Theorems . . . . . . . . . . . . . . . . . 326
11.8 The Continuous Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . 329
11.9 MGFs, CGFs and Weak Convergence . . . . . . . . . . . . . . . . . . . . . 330
11.10 The Central Limit Theorem for Triangular Arrays . . . . . . . . . . . . . 332
11.11 Weak Convergence of Random Vectors . . . . . . . . . . . . . . . . . . . . 334
11.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
12 Large Sample Estimation — Basic Principles 341
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
12.2 The δ-Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
12.3 Variance Stabilizing Transformations . . . . . . . . . . . . . . . . . . . . . 344
12.4 The δ-Method and Higher-Order Approximations . . . . . . . . . . . . . . 347
12.5 The Multivariate δ-Method . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.6 Approximating the Distributions of Sample Quantiles: The Bahadur
Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
12.7 A Central Limit Theorem for U-statistics . . . . . . . . . . . . . . . . . . 357
12.8 The Information Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 359
viii Contents
12.9 Asymptotic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
12.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
13 Asymptotic Theory for Estimating Equations 371
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
13.2 Consistency and Asymptotic Normality of M-Estimators . . . . . . . . . 373
13.3 Asymptotic Theory of MLEs . . . . . . . . . . . . . . . . . . . . . . . . . 375
13.4 A General Form for Regression Models . . . . . . . . . . . . . . . . . . . . 376
13.5 Nonlinear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
13.6 Generalized Linear Models (GLM) . . . . . . . . . . . . . . . . . . . . . . 379
13.7 Generalized Estimating Equations (GEE) . . . . . . . . . . . . . . . . . . 385
13.8 Existence and Consistency of M-Estimators . . . . . . . . . . . . . . . . . 387
13.9 Asymptotic Distribution ofθθˆθ . . . . . . . . . . . . . . . . . . . . . . . . . 390
n
13.10 Regularity Conditions for Estimating Equations . . . . . . . . . . . . . . . 391
13.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
14 Large Sample Hypothesis Testing 395
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
14.2 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
14.3 Large Sample Tests for Simple Null Hypotheses . . . . . . . . . . . . . . . 397
14.4 Nuisance Parameters and Composite Null Hypotheses . . . . . . . . . . . 403
14.5 Pearson’s χ2 Test for Independence in Contingency Tables . . . . . . . . . 408
14.6 A Comparison of the LR, Wald and Score Tests . . . . . . . . . . . . . . . 409
14.7 Confidence Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
14.8 Estimating Power for Approximate χ2 Tests . . . . . . . . . . . . . . . . . 411
14.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
A Parametric Classes of Densities 415
B Topics in Linear Algebra 417
B.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
B.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
B.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
B.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
B.5 Dimension of a Subset of Rd . . . . . . . . . . . . . . . . . . . . . . . . . . 425
C Topics in Real Analysis and Measure Theory 427
C.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
C.2 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
C.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
C.4 Exchange of Integration and Differentiation . . . . . . . . . . . . . . . . . 430
C.5 The Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . 432
C.6 Stirling’s Approximation of the Factorial . . . . . . . . . . . . . . . . . . . 432
C.7 The Gradient Vector and the Hessian Matrix . . . . . . . . . . . . . . . . 432
C.8 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
C.9 Taylor’s Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 435
Contents ix
D Group Theory 437
D.1 Definition of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
D.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
D.3 Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
D.4 Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
D.5 Orbits and Maximal Invariants . . . . . . . . . . . . . . . . . . . . . . . . 442
Bibliography 445
Index 453