Extended empirical likelihood

经验似然方法(Owen，2001)是一种强大的非参数统计推断方法，有着广泛的应用。然而，经验似然置信域存在覆盖不足的问题，因为其覆盖概率往往低于名义水平。这个问题在小样本和多层面的情况下尤为严重。这部分是由于经验似然统计量收敛到极限卡方随机变量的速度，部分是由于经验似然公式中嵌入的凸壳约束(Tsao，2013)。现有的欠覆盖问题的求解方法大致可以分为两类：一类是以提高收敛速度为目标的方法，另一类是以凸包约束为目标的方法。Tsao(2013)和Tsao and Wu(2013)的扩展经验似然属于后者。它的动机是几何扩展原始的经验似然置信域，同时保持其数据驱动的形状。*本建议的主要目的是通过扩展经验似然方法的关键组成部分--基础合成相似映射的扩展因子来深入研究扩展的经验似然方法，以加强其理论基础，进一步提高其已经令人印象深刻的精度。为了实现这一目标，我的研究将集中于通过了解最优扩展因子对样本大小、基础分布的高阶矩和参数向量的维度的依赖来确定最优扩展因子。*本建议的次要目标是研究几个与扩展经验似然相关的项目。这些是[1]发现这种方法的新应用，[2]简化了它的理论和计算，[3]研究了当数据的维度随着样本量增加时它的渐近行为。*这项研究的影响将是巨大的。与主要目标相关的结果将使扩展经验似然法建立在坚实的理论基础上，并显著提高其准确性。关于次级目标的结果将[i]为以前使用原始经验似然的经验似然应用带来更准确的推断，[ii]使该方法更易于使用并适用于更广泛的问题，以及[iii]使该方法能够应用于高维数据，这是大数据分析中的一个重要主题。**

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.**