Akkerboom Testing Problems with Linear or Angular Inequality Constraints

1990
ISBN: 978-1-4612-3392-3
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 62, 291 Seiten, eBook

Reihe: Lecture Notes in Statistics

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).

Zielgruppe

Research

Autoren/Hrsg.

Akkerboom, Johan C.

Inhaltsverzeichnis

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 andangle 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 Mokken’s latent trait model.- References and Author Index.- Appendices.