Adaptive Critical Value for Constrained Likelihood Ratio Testing
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We present a new way of testing ordered hypotheses against all alternatives which overpowers the classical approach both in simplicity and statistical power. Our new method tests the constrained likelihood ratio statistic against the quantile of one and only one chi-squared random variable chosen according to the data instead of a mixture of chi-squares. Our new test is proved to have a valid significance level α and provides more power especially for sparse alternatives in comparison to the classical approach. Our method is also easier to use than the classical approach which needs to calculate a quantile of a mixture of chi-squared random variables with weights that are generally very difficult to calculate. Two cases are considered with details, the case of testing μ_1<0, ..., μ_n<0 and the isotonic case of testing μ_1<μ_2<μ_3 against all alternatives. Contours of the difference in power are shown for these examples showing the interest of our new approach. When the alternative hypothesis is an ordered hypothesis (the null is only a linear space), our new approach applies as well providing an easy test. However, our new approach becomes more powerful than the classical approach only in adjacent areas of the alternative.
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