Constrained statistical inference : inequality, order and shape restrictions

Statistical inference is finding useful applications in numerous fields, from sociology and econometrics to biostatistics. This volume enables professionals in these and related fields to master the concepts of statistical inference under inequality constraints and to apply the theory to problems in a variety of areas.

Constrained Statistical Inference: Order, Inequality, and Shape Constraints provides a unified and up-to-date treatment of the methodology. It clearly illustrates concepts with practical examples from a variety of fields, focusing on sociology, econometrics, and biostatistics.

The authors also discuss a broad range of other inequality-constrained inference problems that do not fit well in the contemplated unified framework, providing a meaningful way for readers to comprehend methodological resolutions.

Chapter coverage includes:

Population means and isotonic regression Inequality-constrained tests on normal means Tests in general parametric models Likelihood and alternatives Analysis of categorical data Inference on monotone density function, unimodal density function, shape constraints, and DMRL functions Bayesian perspectives, including Stein's Paradox, shrinkage estimation, and decision theory

Author Notes

MERVYN J. SILVAPULLE, PhD, is an Associate Professor in the Department of Statistical Science at La Trobe University in Bundoora, Australia. He received his PhD in statistics from the Australian National University in 1981.

PRANAB K. SEN, PhD, is a Professor in the Departments of Biostatistics and Statistics and Operations Research at the University of North Carolina at Chapel Hill. He received his PhD in 1962 from Calcutta University, India.

Dedication	p. v
Preface	p. xv
1 Introduction	p. 1
1.1 Preamble	p. 1
1.2 Examples	p. 2
1.3 Coverage and Organization of the Book	p. 23
2 Comparison of Population Means and Isotonic Regression	p. 25
2.1 Ordered Alternative Hypotheses	p. 27
2.2 Ordered Null Hypotheses	p. 38
2.3 Isotonic Regression	p. 42
2.4 Isotonic Regression: Results Related to Computational Formulas	p. 46
2.5 Appendix: Proofs	p. 53
Problems	p. 57
3 Tests on Multivariate Normal Mean	p. 59
3.1 Introduction	p. 59
3.2 Statement of Two General Testing Problems	p. 60
3.3 Theory: The Basics in Two Dimensions	p. 63
3.4 Chi-bar-square Distribution	p. 75
3.5 Computing the Tail Probabilities of Chi-bar-square Distributions	p. 78
3.6 Results on Chi-bar-square Weights	p. 81
3.7 LRT for Type A problems: V is Known	p. 83
3.8 LRT for Type B problems: V is Known	p. 90
3.9 Tests on the Linear Regression Parameter	p. 95
3.10 Tests When V is Unknown (Perlman's Test and Alternatives)	p. 100
3.11 Optimality Properties	p. 107
3.12 Appendix 1: Convex Cones, Polyhedrals, and Projections	p. 111
3.13 Appendix 2: Proofs	p. 125
Problems	p. 133
4 Tests in General Parametric Models	p. 143
4.1 Introduction	p. 143
4.2 Preliminaries	p. 145
4.3 Tests of R[theta] = 0 Against R[theta greater than or equal] 0	p. 148
4.4 Tests of h([theta]) = 0	p. 164
4.5 An Overview of Score Tests with no Inequality Constraints	p. 168
4.6 Local Score-type Tests of H[subscript 0] : [psi] = 0 Against H[subscript 1] : [psi set membership Psi]	p. 175
4.7 Approximating Cones and Tangent Cones	p. 183
4.8 General Testing Problems	p. 194
4.9 Properties of the mle When the True Value is on the Boundary	p. 209
4.10 Appendix: Proofs	p. 215
5 Likelihood and Alternatives	p. 221
5.1 Introduction	p. 221
5.2 The Union-Intersection Principle	p. 222
5.3 Intersection Union Tests (IUT)	p. 235
5.4 Nonparametrics	p. 243
5.5 Restricted Alternatives and Simes-type Procedures	p. 264
5.6 Concluding Remarks	p. 275
Problems	p. 276
6 Analysis of Categorical Data	p. 283
6.1 Introduction	p. 283
6.2 Motivating Examples	p. 285
6.3 Independent Binomial Samples	p. 292
6.4 Odds Ratios and Monotone Dependence	p. 298
6.5 Analysis of 2 x c contingency tables	p. 306
6.6 Test to Establish that Treatment is Better Than Control	p. 313
6.7 Analysis of r x c Tables	p. 315
6.8 Square Tables and Marginal Homogeneity	p. 322
6.9 Exact Conditional Tests	p. 324
6.10 Discussion	p. 335
6.11 Proofs	p. 335
Problems	p. 338
7 Beyond Parametrics	p. 345
7.1 Introduction	p. 345
7.2 Inference on Monotone Density Function	p. 346
7.3 Inference on Unimodal Density Function	p. 354
7.4 Inference on Shape-Constrained Hazard Functionals	p. 357
7.5 Inference on DMRL Functions	p. 362
7.6 Isotonic Nonparametric Regression: Estimation	p. 366
7.7 Shape Constraints: Hypothesis Testing	p. 369
Problems	p. 374
8 Bayesian Perspectives	p. 379
8.1 Introduction	p. 379
8.2 Statistical Decision Theory Motivations	p. 380
8.3 Stein's Paradox and Shrinkage Estimation	p. 384
8.4 Constrained Shrinkage Estimation	p. 388
8.5 PCC and Shrinkage Estimation in CSI	p. 396
8.6 Bayes Tests in CSI	p. 400
8.7 Some Decision Theoretic Aspects: Hypothesis Testing	p. 402
Problems	p. 404
9 Miscellaneous Topics	p. 407
9.1 Two-sample Problem with Multivariate Responses	p. 408
9.2 Testing that an Identified Treatment is the Best: the Min Test	p. 422
9.3 Cross-over Interaction	p. 434
9.4 Directed Tests	p. 455
Problems	p. 463
Bibliography	p. 469
Index	p. 525

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