Constrained Statistical Inference: Order, Inequality, and Shape Constraints

Preface.

1. Introduction.

1.1 Preamble.

1.2 Examples.

1.3 Coverage and Organization of the Book.

2. Comparison of Population Means and Isotonic Regression.

2.1 Ordered Hypothesis Involving Population Means.

2.2 Test of Inequality Constraints.

2.3 Isotonic Regression.

2.4 Isotonic Regression: Results Related to Computational Formulas.

3. Two Inequality Constrained Tests on Normal Means.

3.1 Introduction.

3.2 Statement of Two General Testing Problems.

3.3 Theory: The Basics in 2 Dimensions.

3.4 Chi-bar-square Distribution.

3.5 Computing the Tail Probabilities of chi-bar-square Distributions.

3.6 Detailed Results relating to chi-bar-square Distributions.

3.7 LRT for Type A Problems: V is known.

3.8 LRT for Type B Problems: V is known.

3.9 Inequality Constrained Tests in the Linear Model.

3.10 Tests When V is known.

3.11 Optimality Properties.

3.12 Appendix 1: Convex Cones.

3.13 Appendix B. Proofs.

4. Tests in General Parametric Models.

4.1 Introduction.

2.2 Preliminaries.

4.3 Tests of Rθ = 0 against Rθ ≥ 0.

4.4 Tests of h(θ) = 0.

4.5 An Overview of Score Tests with no Inequality Constraints.

4.6 Local Score-type Tests of H_o : ψ = 0 vs H₁ : ψ &epsis; Ψ.

4.7 Approximating Cones and Tangent Cones.

4.8 General Testing Problems.

4.9 Properties of the mle When the True Value is on the Boundary.

5. Likelihood and Alternatives.

5.1 Introduction.

5.2 The Union-Intersection principle.

5.3 Intersection Union Tests (IUT).

5.4 Nanparametrics.

5.5 Restricted Alternatives and Simes-type Procedures.

5.6 Concluding Remarks.

6. Analysis of Categorical Data.

6.1 Motivating Examples.

6.2 Independent Binomial Samples.

6.3 Odds Ratios and Monotone Dependence.

6.4 Analysis of 2 x c Contingency Tables.

6.5 Test to Establish that Treatment is Better than Control.

6.6 Analysis of r x c Tables.

6.7 Square Tables and Marginal Homogeneity.

6.8 Exact Conditional Tests.

6.9 Discussion.

7. Beyond Parametrics.

7.1 Introduction.

7.2 Inference on Monotone Density Function.

7.3 Inference on Unimodal Density Function.

7.4 Inference on Shape Constrained Hazard Functionals.

7.5 Inference on DMRL Functions.

7.6 Isotonic Nonparametric Regression: Estimation.

7.7 Shape Constraints: Hypothesis Testing.

8. Bayesian Perspectives.

8.1 Introduction.

8.2 Statistical Decision Theory Motivations.

8.3 Stein’s Paradox and Shrinkage Estimation.

8.4 Constrained Shrinkage Estimation.

8.5 PC and Shrinkage Estimation in CSI.

8.6 Bayes Tests in CSI.

8.7 Some Decision Theoretic Aspects: Hypothesis Testing.

9. Miscellaneous Topics.

9.1 Two-sample Problem with Multivariate Responses.

9.2 Testing that an Identified Treatment is the Best: The mini-test.

9.3 Cross-over Interaction.

9.4 Directed Tests.

Bibliography.

Index.