Constrained Statistical Inference

An up-to-date approach to understanding statistical inference 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

An up-to-date approach to understanding statistical inferenceStatistical inference is finding useful applications in numerousfields, from sociology and econometrics to biostatistics. Thisvolume enables professionals in these and related fields to masterthe concepts of statistical inference under inequality constraintsand to apply the theory to problems in a variety of areas.Constrained Statistical Inference: Order, Inequality, and ShapeConstraints provides a unified and up-to-date treatment of themethodology. It clearly illustrates concepts with practicalexamples from a variety of fields, focusing on sociologyeconometrics, and biostatistics.The authors also discuss a broad range of otherinequality-constrained inference problems that do not fit well inthe contemplated unified framework, providing a meaningful way forreaders 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 densityfunction, shape constraints, and DMRL functions* Bayesian perspectives, including Stein's Paradoxshrinkage estimation, and decision theory

Dedication.Preface.1. Introduction.1.1 Preamble.1.2 Examples.1.3 Coverage and Organization of the Book.2. Comparison of Population Means and IsotonicRegression.2.1 Ordered Hypothesis Involving Population Means.2.2 Test of Inequality Constraints.2.3 Isotonic Regression.2.4 Isotonic Regression: Results Related to ComputationalFormulas.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-squareDistributions.3.6 Detailed Results relating to chi-bar-squareDistributions.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 Rtheta = 0 against Rtheta >= 0.4.4 Tests of h(theta) = 0.4.5 An Overview of Score Tests with no InequalityConstraints.4.6 Local Score-type Tests of Ho : psi = 0 vsH1 : psi &epsis; Psi.4.7 Approximating Cones and Tangent Cones.4.8 General Testing Problems.4.9 Properties of the mle When the True Value is on theBoundary.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: Themini-test.9.3 Cross-over Interaction.9.4 Directed Tests.Bibliography.Index.