On testing a class of restricted hypotheses

The aim of this paper is to provide the asymptotic distribution function, under the null and the alternative hypothesis, of statistics based on the likelihood, when the parameter space is restricted by a cone C such that D C is a spherical cone or the polyhedral cone { t ∈ R N + 1 : t 1 ⩾ 0 , … , t...

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Bibliographic Details
Published in:Journal of statistical planning and inference 2007-04, Vol.137 (4), p.1343-1361
Main Authors: Delmas, Céline, Foulley, Jean-Louis
Format: Article
Language:English
Subjects:
Cone
Exact sciences and technology
General topics
Likelihood ratio test
Mathematics
Multivariate analysis
Nonparametric inference
Parametric inference
Probability and statistics
Projection
Restricted hypothesis
Rice method
Sciences and techniques of general use
Score test
Statistics
Statistics Theory
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Summary:The aim of this paper is to provide the asymptotic distribution function, under the null and the alternative hypothesis, of statistics based on the likelihood, when the parameter space is restricted by a cone C such that D C is a spherical cone or the polyhedral cone { t ∈ R N + 1 : t 1 ⩾ 0 , … , t N + 1 ⩾ 0 } where D is a fixed known positive definite matrix. The results obtained ensure the calculation of threshold and power of such restricted tests. Our study is not limited to i.i.d. observations. Numerical calculations show that the likelihood ratio test restricted by a cone C 1 ⊂ C 2 is uniformly more powerful than the likelihood ratio test restricted by C 2 on all the alternatives that belong to C 1 . Particularly, a restricted test is uniformly more powerful than a non-restricted one on all the restricted alternatives.
ISSN:0378-3758
1873-1171
DOI:10.1016/j.jspi.2006.04.006