Empirical Likelihood With Applications

Empirical likelihood, first introduced by Thomas and Grunkemeier (1975) and later extended in Owen (1988, 1990), is an effective and flexible nonparametric method based on a data-driven likelihood ratio function. It enjoys many advantages over other nonparametric methods, such as automatic determination of the confidence region by the sample and transformation respecting, easy incorporation of side information, direct extension to biased sampling and censored data, good asymptotic power properties and Bartlete correctability. The empirical likelihood method can be used to find estimators, conduct hypothesis testing and construct small confidence intervals/regions. However, when treating with nonlinear statistics via the empirical likelihood method, the computation burden is quite heavy. The Jackknife Empirical Likelihood method, brought out by Jing et al. (2009), is surprisingly easy to cope with nonlinear statistics and largely relieves computation burden. In this thesis, we first apply the jackknife empirical likelihood method to make inference for the Volume Under the ROC Surface (VUS) and the Hypervolume Under the ROC Manifold (HUM) measures, which are straight extensions of the Area Under the The Receiver Operating Characteristic (ROC) curve (AUC) for three-category and multi-category samples respectively. The popularity and importance of VUS and HUM are due to their capability of providing general measures of the differences amongst populations. Another problem in this thesis concerns the compound Poisson sum. Monte Carlo simulations are conducted to assess the performance of the proposed methods in finite samples. Some meaningful real datasets are also analyzed.

Author dc:creator

• WANG XIPING

• Compound Poisson process, Empirical likelihood, Jackknife, ROC surface and manifold, U-statistics