More powerful goodness-of-fit tests for uniform stochastic ordering

The ordinal dominance curve (ODC) is a useful graphical tool to compare two population distributions. These distributions are said to satisfy uniform stochastic ordering (USO) if the ODC for them is star-shaped. A goodness-of-fit test for USO was recently proposed when both distributions are unknown. This test involves calculating the L distance between an empirical estimator of the ODC and its least star-shaped majorant. The least favorable configuration of the two distributions was established so that proper critical values could be determined; i.e., to control the probability of type I error for all star-shaped ODCs. However, the use of these critical values can lead to a conservative test and minimal power to detect certain non-star-shaped alternatives. Two new methods for determining data-dependent critical values are proposed. Simulation is used to show both methods can provide substantial increases in power while still controlling the size of the distance-based test. The methods are also applied to a data set involving premature infants. An R package that implements all tests is provided.