Extended empirical likelihood

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The empirical likelihood method (Owen, 2001) is a powerful non-parametric method of statistical inference with many applications. However, the empirical likelihood confidence region suffers from an under-coverage problem in that its coverage probability tends to be lower than the nominal level. The problem is particularly serious in small sample and multidimensional situations. It is partly due to the rate at which the empirical likelihood statistic converges to the limiting chi-square random variable, and partly due to the convex hull constraint embedded in the formulation of the empirical likelihood (Tsao, 2013). Existing methods for the under-coverage problem can be roughly divided into two types: those aimed at increasing the rate of convergence and those targeting the convex hull constraint. The extended empirical likelihood of Tsao (2013) and Tsao and Wu (2013) is in the latter category. It is motivated by geometrically expanding the original empirical likelihood confidence regions while preserving their data driven shape. It is a leading method for dealing with the under-coverage problem.The primary objective of this proposal is to thoroughly study the extended empirical likelihood method through its key component, the expansion factor of the underlying composite similarity mapping, in order to strengthen its theoretical foundation and further improve its already impressive accuracy. To achieve this objective, my research will focus on identifying the optimal expansion factor through understanding its dependence on the sample size, higher moments of the underlying distribution and the dimension of the parameter vector.The secondary objective of this proposal is to work on several related projects concerning the extended empirical likelihood. These are [1] finding new applications of this method, [2] simplifying its theory and computation, and [3] studying its asymptotic behavior when the dimension of the data increases with the sample size. The impact of this research will be significant. Findings related to the primary objective will put the extended empirical likelihood method on a sound theoretical footing and substantially improve its accuracy. Results concerning the secondary objective will [i] bring more accurate inference to empirical likelihood applications that previously use the original empirical likelihood, [ii] make the method easier to use and applicable to a wider range of problems, and [iii] make it possible to apply the method to high dimensional data which is an important topic in Big Data analysis.