Constrained Statistical Inference: Order, Inequality, and Shape ConstraintsISBN: 9780471208273
532 pages
November 2004

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 uptodate 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 inequalityconstrained 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
 Inequalityconstrained 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
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 Chibarsquare Distribution.
3.5 Computing the Tail Probabilities of chibarsquare Distributions.
3.6 Detailed Results relating to chibarsquare 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 Scoretype Tests of H_{o} : ψ = 0 vs H_{1} : 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 UnionIntersection principle.
5.3 Intersection Union Tests (IUT).
5.4 Nanparametrics.
5.5 Restricted Alternatives and Simestype 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 Twosample Problem with Multivariate Responses.
9.2 Testing that an Identified Treatment is the Best: The minitest.
9.3 Crossover Interaction.
9.4 Directed Tests.
Bibliography.
Index.
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.
"…an invaluable resource for any researcher with interests in constrained problems…it is easy to conclude that any statistical library would be incomplete without it." (Biometrics, December 2005)
"…a valuable source of information for statisticians working in any area…" (Mathematical Reviews, 2005k)