A jackknife empirical likelihood approach to goodness of fit U-statistic testing with side information

Motivated by applications to goodness of fit U-statistics testing, the jackknife empirical likelihood of Jing, et al. (2009) is justified with an alternative approach, and the Wilks theorem for vector U-statistics is proved. This generalizes Owen's empirical likelihood theorem for a vector mean to a vector U-statistics-based mean and includes the jackknife empirical likelihood of U-statistics with side information as a special case. The results are generalized to allow for the constraints to use estimated criteria functions and for the number of constraints to grow with the sample size. The latter is needed to handle naturally occurring nuisance parameters in semiparametric models. The developed theory is applied to derive the empirical-likelihood-based goodness-of-fit tests and confidence sets for U-quantiles with finite many constraints and with growing number of constraints in the Theil estimator based test about the slope in a simple linear regression; for the Wilcoxon signed rank test about symmetry with a unknown center of symmetry; for Kendall's tau and Goodman and Kruskal's Gamma with side information; for the test about independence of two categorical outcomes; for joint confidence sets of variances in a balanced random effects model and for the simplicial depth function with finitely many and growing number of constraints. Some of the proposed jackknife empirical likelihood based goodness of fit tests are asymptotically distribution free. A simulation study is conducted to evaluate the behaviors of the Theil test with a finite number and growing number of constraints.