Adaptive critical value for constrained likelihood ratio testing

Biometrika, vol. 107, pp. 677–688

Abstract

Summary We present a new general method for constrained likelihood ratio testing which, when few constraints are violated, improves upon the existing approach in the literature that compares the likelihood ratio with the quantile of a mixture of chi-squared distributions; the improvement is in terms of both simplicity and power. The proposed method compares the constrained likelihood ratio statistic against the quantile of only one chi-squared random variable with data-dependent degrees of freedom. The new test is shown to have a valid exact significance level $\alpha$. It also has more power than the classical approach against alternatives for which the number of violations is not large. We provide more details for testing a simple order $\mu_1\leqslant\cdots\leqslant\mu_p$ against all alternatives using the proposed approach and give clear guidelines as to when the new method would be advantageous. A simulation study suggests that for testing a simple order, the new approach is more powerful in many scenarios than the existing method that uses a mixture of chi-squared variables. We illustrate the results of our adaptive procedure using real data on the liquidity preference hypothesis.

Authors 4

1. Diaa Al Mohamad corresponding Aachen Department of Biomedical Data Sciences

   Leiden University Medical Center

   Affiliation as printed

   Department of Biomedical Data Sciences, Leiden University Medical Center, Einthovenweg 20, 2333 ZC Leiden, The Netherlands

2. Erik W. van Zwet Aachen Department of Biomedical Data Sciences

   Leiden University Medical Center

   Affiliation as printed

   Department of Biomedical Data Sciences, Leiden University Medical Center, Einthovenweg 20, 2333 ZC Leiden, The Netherlands

3. Eric Cator

   Radboud University Nijmegen

   Affiliation as printed

   Faculty of Sciences, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

4. Jelle J. Goeman Aachen Department of Biomedical Data Sciences

   Leiden University Medical Center

   Affiliation as printed

   Department of Biomedical Data Sciences, Leiden University Medical Center, Einthovenweg 20, 2333 ZC Leiden, The Netherlands