Constrained Statistical Inference: Inequality, Order, and Shape Restrictions

Constrained Statistical Inference: Inequality, Order, and Shape Restrictions......Page 5
Contents......Page 9
Dedication......Page 7
Preface......Page 17
1.1 Preamble......Page 21
1.2 Examples......Page 22
1.3 Coverage and Organization of the Book......Page 43
2 Comparison of Population Means and Isotonic Regression......Page 45
2.1 Ordered Alternative Hypotheses......Page 47
2.1.1 Test of H0 : μ1 = μ2 = μ3 Against an Order Restriction......Page 48
2.1.2 Test of H0 = μ1 = ... = μk Against an Order Restriction......Page 51
2.1.2.1 Exact finite sample results when the error distribution is N(0, σ2)......Page 55
2.1.2.2 Computing minµεH1 Σ Σ(yij – µi)2, F, and E2......Page 56
2.1.3 General Remarks......Page 57
2.2 Ordered Null Hypotheses......Page 58
2.3 Isotonic Regression......Page 62
2.3.1 Quasi-order and Isotonic Regression......Page 64
2.4 Isotonic Regression: Results Related to Computational Formulas......Page 66
2.4.1 Isotonic Regression Under Simple Order......Page 67
2.4.2 Ordered Means of Exponential Family......Page 70
2.5 Appendix: Proofs......Page 73
Problems......Page 77
3.1 Introduction......Page 79
3.2 Statement of Two General Testing Problems......Page 80
3.2.1 Reduction by Sufficiency......Page 82
3.3 Theory: The Basics in Two Dimensions......Page 83
3.4 Chi-bar-square Distribution......Page 95
3.5 Computing the Tail Probabilities of Chi-bar-square Distributions......Page 98
3.6 Results on Chi-bar-square Weights......Page 101
3.7 LRT for Type A problems: V is Known......Page 103
3.8 LRT for Type B problems: V is Known......Page 110
3.9 Tests on the Linear Regression Parameter......Page 115
3.9.1 Null Distributions......Page 117
3.10 Tests When V is Unknown (Perlman's Test and Alternatives)......Page 120
3.10.1 Type A Testing Problem......Page 122
3.10.2 Type B Testing Problem......Page 124
3.10.3 Conditional Tests of H0 : θ = 0 vs H1 : θ >= 0......Page 125
3.11.1 Consistency of Tests......Page 127
3.11.2 Monotonicity of the Power Function......Page 129
3.12.1 Introduction......Page 131
3.12.2 Projections onto Convex Cones......Page 133
3.12.3 Polyhedral Cones......Page 142
3.13 Appendix 2: Proofs......Page 145
Problems......Page 153
4.1 Introduction......Page 163
4.2 Preliminaries......Page 165
4.3.1 Likelihood Ratio Test with iid Observations......Page 168
4.3.2 Tests in the Presence of Nuisance Parameters......Page 170
4.3.3 Wald- and Score-type Tests with iid Observations......Page 173
4.3.4 Independent Observations with Covariates......Page 176
4.3.5 Examples......Page 180
4.4.1 Test of h(θ) = 0 Against h(θ) >= 0......Page 184
4.4.2 Test of h(θ) = 0 Against h2(θ) >= 0......Page 187
4.5 An Overview of Score Tests with no Inequality Constraints......Page 188
4.5.1 Test of H0 : θ = θ0 Against H2 : θ = θ0......Page 189
4.5.2 Test of H0 : φ = φ0 Against H2 : φ = φ0......Page 191
4.5.3 Tests Based on Estimating Equations......Page 192
4.6.1 Local Score Test of H0 : θ = 0 Against H1 : θ ε C......Page 195
4.6.2 Local Score Test of H0 : φ = 0 Against H1 : φ ε C......Page 197
4.6.3 Local Sore-Type Test Based on Estimating Equations......Page 198
4.6.4 A General Local Score-Type Test of H0 : φ = 0......Page 199
4.6.5 A Simple One-Dimensional Test of φ = 0 Against φ ε C......Page 200
4.6.6 A Data Example: Test Against ARCH Effect in an ARCH Model......Page 201
4.7 Approximating Cones and Tangent Cones......Page 203
4.7.1 Chernoff Regularity......Page 206
4.8 General Testing Problems......Page 214
4.8.1 Likelihood Approach When Θ is Open......Page 215
4.8.2 Likelihood Approach When Θ is not Open......Page 217
4.8.3 Tests with General Objective Functions......Page 219
4.8.4 Examples......Page 221
4.8.5 Test of h(θ) >= 0 Against h(θ) > 0......Page 222
4.9 Properties of the mle When the True Value is on the Boundary......Page 229
4.10 Appendix: Proofs......Page 235
5.1 Introduction......Page 241
5.2 The Union-Intersection Principle......Page 242
5.2.1 UIT on Multinomial Mean......Page 244
5.3 Intersection Union Tests (IUT)......Page 255
5.4.1 Two-Sample Problem Revisited......Page 263
5.4.2 Ordered Alternative Problem Revisited......Page 265
5.4.3 Ordered Alternatives: Two-Way Layouts......Page 268
5.4.4 UIP and LMPR Tests for Restricted Alternatives.......Page 273
5.4.5 Nonparametric UIT for Restricted Alternatives......Page 281
5.5 Restricted Alternatives and Simes-type Procedures......Page 284
5.5.1 Probability Integral Transformation (PIT) and OSL or p- Values......Page 287
5.5.2 Simes-Type Procedures......Page 289
5.6 Concluding Remarks......Page 295
Problems......Page 296
6.1 Introduction......Page 303
6.2 Motivating Examples......Page 305
6.3 Independent Binomial Samples......Page 312
6.3.1 Test of H0 : R1π = 0 and R2π = 0 Against H1 : R2π >= 0......Page 313
6.3.3 Test of H0 : Aπ > 0 Against H1 : Aπ > 0......Page 315
6.3.4 Pairwise Contrasts of Binomial Parameters......Page 316
6.3.5 Discussion......Page 317
6.4 Odds Ratios and Monotone Dependence......Page 318
6.4.1 2 × 2 Tables......Page 319
6.4.2 2 × c Tables......Page 320
6.4.3 r × c Tables......Page 323
6.5 Analysis of 2 × C Contingency Tables......Page 326
6.5.1 LRT of π1 = π2 Against Simple Stochastic Order......Page 328
6.5.2 LRT for Simple Stochastic Order: Rπ >= 0 vs Rπ > 0......Page 331
6.6 Test to Establish that Treatment is Better Than Control......Page 333
6.7 Analysis of r × c Tables......Page 335
6.7.1 LRT Against Simple Stochastic Order......Page 337
6.7.2 LRT for and Against Stochastic Orders......Page 340
6.8 Square Tables and Marginal Homogeneity......Page 342
6.9 Exact Conditional Tests......Page 344
6.9.1 Exact Conditional Tests in 2 × c Tables......Page 345
6.9.2 Comparison of Several Binomial Parameters......Page 347
6.9.3 Exact Conditional Tests in r × c Tables......Page 348
6.9.4 Marginal Homogeneity in Square Tables (Agresti-Coull test)......Page 351
6.11 Proofs......Page 355
Problems......Page 358
7.1 Introduction......Page 365
7.2 Inference on Monotone Density Function......Page 366
7.3 Inference on Unimodal Density Function......Page 374
7.4 Inference on Shape-Constrained Hazard Functionals......Page 377
7.5 Inference on DMRL Functions......Page 382
7.6 Isotonic Nonparametric Regression: Estimation......Page 386
7.7 Shape Constraints: Hypothesis Testing......Page 389
Problems......Page 394
8.1 Introduction......Page 399
8.2 Statistical Decision Theory Motivations......Page 400
8.3 Stein's Paradox and Shrinkage Estimation......Page 404
8.4 Constrained Shrinkage Estimation......Page 408
8.5 PCC and Shrinkage Estimation in CSI......Page 416
8.6 Bayes Tests in CSI......Page 420
8.7 Some Decision Theoretic Aspects: Hypothesis Testing......Page 422
Problems......Page 424
9 Miscellaneous Topics......Page 427
9.1 Two-sample Problem with Multivariate Responses......Page 428
9.1.1 One Degree of Freedom Tests of H0 : µ1 – µ2 = 0 Against H1 : µ1 – µ2 > 0......Page 431
9.1.2 O'Brien's Test of H0 : µ1 – µ2 = 0 vs H1 : µ1 – µ2 >= 0......Page 432
9.1.3 A One Degree of Freedom Test for the Equality of General Parameters......Page 434
9.1.4.1 X1 ~ N(µ1, σ2U1) and X2 ~ N(µ2, σ2U2)......Page 436
9.1.4.2 X1 ~ N(µ1, σ12U1) and X2 ~ N(µ2, σ22U2)......Page 437
9.1.4.4 An extension of Follmann's test......Page 438
9.1.5 Hi : New Treatment is Noninferior With Respect to Every Variable and Superior With Respect to Some Variables......Page 440
9.2 Testing that an Identified Treatment is the Best: the Min Test......Page 442
9.2.1 Monotone Critical Regions and Tests......Page 448
9.2.2 Optimality of Min Test......Page 449
9.2.3 Multivariate Responses......Page 450
9.2.4 Comparison of LRT with the Min Test......Page 451
9.3 Cross-over Interaction......Page 454
9.3.1 A Test Against Cross-over Interaction: Two Treatments and I Groups.......Page 457
9.3.1.1 Case 1: δi ~ N(δi, σi2) where σi2 = σ2wi, σ2 is unknown and wi is known, i = 1..., I......Page 459
9.3.1.2 Case 2: δi ~ N(δi, σi2), δ1,..., δ1 are independent, and σ1,..., σ1 are known......Page 462
9.3.1.3 Case 3: δi is approximately N(δi, σi2) and δi is unknown, i = 1,..., I......Page 463
9.3.2 Robustness Against Outliers......Page 464
9.3.3 Dependent Treatment Effects......Page 465
9.3.4 A Range Test Against Cross-over Interaction......Page 467
9.3.5 A Test with Uniformly More Power and Less Bias......Page 468
9.3.6 Cross-over Interaction of Practical Significance......Page 469
9.3.7 More than Two Treatments......Page 471
9.3.8 H0: Cross-over Interaction is Present......Page 472
9.3.9 Proof of Proposition 9.3.1......Page 473
9.4 Directed Tests......Page 475
9.4.1 Directed Test in a Simple Case......Page 476
9.4.2 Directed Test Against Stochastic Order......Page 478
9.4.3 r x c Tables......Page 480
9.4.4 Improving Power by Enlarging the Achievable Set of p-Values......Page 482
Problems......Page 483
Bibliography......Page 489
Index......Page 545