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Constrained Statistical Inference: Order, Inequality, and Shape Constraints

Constrained Statistical Inference: Order, Inequality, and Shape Constraints

Mervyn J. Silvapulle, Pranab Kumar Sen

ISBN: 978-0-471-20827-3

Nov 2004

532 pages

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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

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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 Ho : 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.



"This monograph provides an excellent coverage of the last twenty years of constrained statistical inference." (Journal of the American Statistical Association, March 2006)

"…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)