Standard empirical likelihood for U-statistics is too computationally expensive due to the nonlinear constraint in the underlying optimization problem. To sidestep this difficult computational issue, the pseudo-values empirical likelihood method is proposed in this thesis. Motivated by the fact that the jackknife pseudo-values are asymptotically independent and identically distributed, we apply the standard empirical likelihood to the mean functional based on these jackknife pseudo-values although they are dependent in general. Wilks's theorem is shown under the second moment condition, which may be used to construct confidence intervals or do hypothesis testing....[ Read more ]

Standard empirical likelihood for U-statistics is too computationally expensive due to the nonlinear constraint in the underlying optimization problem. To sidestep this difficult computational issue, the pseudo-values empirical likelihood method is proposed in this thesis. Motivated by the fact that the jackknife pseudo-values are asymptotically independent and identically distributed, we apply the standard empirical likelihood to the mean functional based on these jackknife pseudo-values although they are dependent in general. Wilks's theorem is shown under the second moment condition, which may be used to construct confidence intervals or do hypothesis testing.

This method of combining jackknife and empirical likelihood could work more generally than just for U-statistics. We also make an attempt to apply the pseudo-values empirical likelihood to generalized U-statistics. Wilks's theorem is shown to hold under mild conditions after a long mathematical proof. The pseudo-values empirical likelihood for two sample U-statistics is extremely simple to use, and yet has very good coverage properties from our simulation study. As applications, we make statistical inference about P(X < Y), the so-called stress-strength model, and study the efficiency of different diagnostic markers via comparing the areas under the ROC curves of markers.