An improved bootstrap test of density ratio ordering

Two probability distributions with common support are said to exhibit density ratio ordering when they admit a nonincreasing density ratio. Existing statistical tests of the null hypothesis of density ratio ordering are known to be conservative, with null limiting rejection rates below the nominal significance level whenever the two distributions are unequal. We show how a bootstrap procedure can be used to shrink the critical values used in existing procedures such that the limiting rejection rate is increased to the nominal significance level on the boundary of the null. This improves power against nearby alternatives. Our procedure is based on preliminary estimation of a contact set, the form of which is obtained from a novel representation of the Hadamard directional derivative of the least concave majorant operator. Numerical simulations indicate that improvements to power can be very large in moderately sized samples. We thank Zhonglin Li and Juwon Seo for research assistance, and Andres Santos and seminar participants at the University of Texas at Austin, Hong Kong University of Science and Technology, University of Tokyo, University of Sydney, Pennsylvania State University, University College London, University of Copenhagen, Aarhus University, and Hitotsubashi University for helpful comments.