Nonparametric tests for and against likelihood ratio ordering in the two-sample problem

We derive nonparametric procedures for testing for and against likelihood ratio ordering in the two-population setting with continuous distributions. We account for this ordering by examining the least concave majorant of the ordinal dominance curve formed from the nonparametric maximum likelihood estimators of the continuous distribution functions F and G. In particular, we focus on testing equality of F and G versus likelihood ratio ordering and testing for a violation of likelihood ratio ordering. For both testing problems, we propose area-based and sup-norm-based test statistics, derive appropriate limiting distributions, and provide simulation results that characterise the performance of our procedures. We illustrate our methods using data from a controlled experiment involving the effects of radiation on mice.