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。这是解决覆盖率不足问题的主要方法。

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.