Statistical Inference on the Monotonic Index Model Based on Monotone Rank Estimate

In this paper, we propose a likelihood ratio test statistic for the monotone rank estimators on the monotonic linear index model. Unlike the usual likelihood ratio test, we treat the U-statistic object function as a likelihood, and different from the traditional asymptotic results for likelihood ratio test, we prove that the limiting distribution is a weighted sum of chi-squared istribution, with weights depend on the unknown parameters. To handle this obstacle in practical situation, we apply two methods to approximate the null distribution, the first one is to use a scale-shifted chi-squared approximation, the second one is to use the Rao-Scott correction. We find that both methods work well. A real data is used to illustrate its use in practical situation