Extended empirical likelihood

扩展的经验可能性

• 批准号: RGPIN-2016-03804
• 负责人: Tsao, Min
• 金额: $ 2.4万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759152
• 关键词: Extended; empirical; likelihood

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.

经验似然法（Owen，2001）是一种强大的非参数统计推断方法，具有许多应用。然而，经验似然置信区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。在小样本和多维情况下这个问题尤其严重。部分原因在于经验似然统计量收敛到限制卡方随机变量的速率，部分原因在于经验似然公式中嵌入的凸包约束（Tsao，2013）。现有的解决覆盖不足问题的方法大致可以分为两类：旨在提高收敛速度的方法和针对凸包约束的方法。 Tsao (2013) 以及 Tsao 和 Wu (2013) 的扩展经验似然属于后一类。它的动机是几何扩展原始经验似然置信区域，同时保留其数据驱动的形状。它是处理覆盖不足问题的领先方法。该提案的主要目标是通过其关键组成部分（底层复合相似映射的扩展因子）深入研究扩展经验似然方法，以加强其理论基础基础并进一步提高其已经令人印象深刻的准确性。为了实现这一目标，我的研究将侧重于通过了解最佳扩展因子对样本大小、基础分布的高矩以及参数向量的维度的依赖性来确定最佳扩展因子。该提案的第二个目标是研究几个相关的问题有关扩展经验可能性的项目。这些是[1]寻找该方法的新应用，[2]简化其理论和计算，以及[3]研究当数据维数随着样本大小增加时其渐近行为。这项研究的影响将是巨大的。与主要目标相关的发现将为扩展经验似然法奠定坚实的理论基础，并大幅提高其准确性。有关次要目标的结果将 [i] 为先前使用原始经验似然的经验似然应用带来更准确的推论，[ii] 使该方法更易于使用并适用于更广泛的问题，并且 [iii] 使其成为可能将该方法应用于高维数据，这是大数据分析中的一个重要课题。