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AuthorAkkerboom, Johan C. author
TitleTesting Problems with Linear or Angular Inequality Constraints [electronic resource] / by Johan C. Akkerboom
ImprintNew York, NY : Springer New York, 1990
Connect tohttp://dx.doi.org/10.1007/978-1-4612-3392-3
Descript XII, 291 p. 1 illus. online resource

SUMMARY

Represents a self-contained account of a new promising and generally applicable approach to a large class of one-sided testing problems, where the alternative is restricted by at least two linear inequalities. It highlights the geometrical structure of these problems. It gives guidance in the construction of a so-called Circular Likelihood Ratio (CLR) test, which is obtained if the linear inequalities, or polyhedral cone, are replaced by one suitable angular inequality, or circular cone. Such a test will often constitute a nice and easy-to-use compromise between the LR-test and a suitable linear test against the original alternative. The book treats both theory and practice of CLR-tests. For cases with up to 13 linear inequalities, it evaluates the power of CLR-tests, derives the most stringent CLR-test, and provides tables of critical values. It is of interest both to the specialist in order- restricted inference and to the statistical consultant in need of simple and powerful one-sided tests. Many examples are worked out for ANOVA, goodness-of-fit, and contingency table problems. Case studies are devoted to Mokken's one- dimensional scaling model, one-sided treatment comparison in a two-period crossover trial, and some real data ANOVA- layouts (biology and educational psychology)


CONTENT

1 Testing problems with linear inequality constraints -- 1.0 General introduction and outline of results -- 1.1* Notations -- 1.2 Testing statistical hypotheses -- 1.3 Cases from statistical practice -- 1.4 The general problem with the alternative restricted by linear inequalities -- 1.5 The canonical form: testing against the pointed polyhedral cone K -- 1.6 Particular classes of testing problems with the alternative restricted by linear inequalities -- 1.7 Problems with the null hypothesis restricted by linear inequalities -- 2 The main problem: testing against the pointed polyhedral cone K -- 2.0 Introduction and summary -- 2.1* Linear inequality constraints and the geometry of polyhedral cones -- 2.2 Linear tests -- 2.3 Likelihood ratio tests -- 2.4 Testing a polyhedral-cone-shaped null hypothesis -- 3 A modification of the main problem: testing against a circular cone -- 3.0 Introduction and summary -- 3.1* An angular inequality constraint and the geometry of circular cones -- 3.2 Likelihood ratio tests for the modified problem -- 3.3* Computation of critical values of the likelihood ratio test statistics for the modified problem -- 3.4 A reduction of the modified problem by sufficiency and invariance -- 3.5 Easy-to-use combination procedures for the reduced modified problem -- 3.6* Other procedures for the reduced modified problem (?2=1) -- 3.7 Some theory about the power properties of invariant tests (?2=1) -- 3.8 Testing a circular-cone-shaped null hypothesis -- 4 Circular likelihood ratio tests for the main problem -- 4.0 Introduction and summary -- 4.1 Replacing the polyhedral cone K by some circular cone -- 4.2* Computation of the power of circular likelihood ratio (CLR-) tests (?2=1) -- 4.3 Minimization of the maximum shortcoming of CLR-tests over K (?2=1) -- 4.4 The minimax ray and angle of K for some particular cases -- 4.5 The maximin ray and angle of K for some particular cases -- 4.6 The use of CLR-tests -- 4.7 Power comparisons -- 4.8 Graphs of the minimax angle and the maximin angle of K for some particular cases -- 5 Applications -- 5.1 One-sided treatment comparison in the two-period crossover trial with binary outcomes -- 5.2 Test expectancy in educational psychology -- 5.3 Predatory behavior of hungry beetles -- 5.4 The assumption of double monotony in Mokkenโ{128}{153}s latent trait model -- References and Author Index -- Appendices


Mathematics Applied mathematics Engineering mathematics Mathematics Applications of Mathematics



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