Nonparametric goodness-of-fit tests for uniform stochastic ordering

Abstract

We propose $L^{p}$ distance-based goodness-of-fit (GOF) tests for uniform stochastic ordering with two continuous distributions $F$ and $G$, both of which are unknown. Our tests are motivated by the fact that when $F$ and $G$ are uniformly stochastically ordered, the ordinal dominance curve $R=FG^{-1}$ is star-shaped. We derive asymptotic distributions and prove that our testing procedure has a unique least favorable configuration of $F$ and $G$ for $p\in [1,\infty]$. We use simulation to assess finite-sample performance and demonstrate that a modified, one-sample version of our procedure (e.g., with $G$ known) is more powerful than the one-sample GOF test suggested by Arcones and Samaniego [Ann. Statist. 28 (2000) 116–150]. We also discuss sample size determination. We illustrate our methods using data from a pharmacology study evaluating the effects of administering caffeine to prematurely born infants.