Empirical likelihood, smoothed likelihood, and their applications

经验似然、平滑似然等非参数似然方法在许多领域都是强有力的工具。它们是有效的统计推断的强大框架，不会对数据强加参数模型。这对于具有复杂结构的数据特别有用。然而，为来自复杂科学环境的数据提出适当的非参数似然方法并研究相应的理论性质可能是非常具有挑战性的。在这一领域有许多未解决的问题。我的研究是关于经验似然和平滑似然方法，以及它们在各种科学数据中的应用。这项提议包括两个主题。 第一个主题是对封闭种群中物种丰富度的估计。这是许多科学领域的热门问题，如生物学、生态学和渔业研究。我将在半参数设置下考虑经验似然方法。我计划研究丰度的最大经验似然估计的渐近性质，并推导出经验似然比检验的极限分布，它将被用来构造丰度的可信区间。这个可信区间至少有两个优点：(1)它的形状是由数据驱动的，不需要是对称的；(2)我们不需要估计丰度估计器的方差。该项目将对物种丰富度的估计和经验似然的理论和方法作出根本性的贡献。 第二个主题是使用最大平滑似然估计混合模型中的未知参数或函数，并提供关于混合比例的辅助信息。这项工作的动机来自许多领域的科学问题，如疟疾研究、经济学研究和不符合规定的随机试验。我将提出一种类似EM的算法来数值计算未知参数和函数的最大平滑似然估计。我将研究算法的收敛和最大平滑似然估计的渐近性质。这一研究方案不仅将解决人们感兴趣的问题，而且有望推动这些方法在其他研究领域的应用。 本提案中的项目旨在解决各个研究领域中具有挑战性的问题。他们将使用先进的统计工具，如混合模型、经验似然和平滑似然来解决困难的科学问题。这项研究计划将反过来有利于对这些方法的理论和方法研究。这两个项目的成果将适用于与NSERC相关的许多领域，如上文概述的领域。

Nonparametric likelihood methods such as the empirical likelihood and the smoothed likelihood are powerful tools in many areas. They are powerful frameworks for valid statistical inference that do not impose a parametric model on the data. This is particularly useful for data with a complicated structure. However, proposing appropriate nonparametric likelihood methods for data from complex scientific settings and studying the corresponding theoretical properties can be very challenging. There are many open-ended problems in this area. My research is concerned with the empirical likelihood and smoothed likelihood methods, and their application to scientific data of various kinds. This proposal consists of two topics. The first topic is the estimation of species abundance in a closed population. This is a popular problem in many scientific areas, such as biology, ecology, and fishery studies. I will consider the empirical likelihood method in a semiparametric setup. I plan to investigate the asymptotic properties of the maximum empirical likelihood estimator of the abundance and to derive the limiting distribution of the empirical likelihood ratio test, which will be used to construct the confidence interval for the abundance. This confidence interval has at least two advantages: (1) its shape is data-driven and does not need to be symmetric; (2) we do not need to estimate the variance of the estimator for the abundance. This project will make a fundamental contribution to the estimation of species abundance and to the theory and methodology of empirical likelihood. The second topic is the use of the maximum smoothed likelihood to estimate the unknown parameters or functions in mixture models with auxiliary information on the mixing proportions. The motivation for this work comes from scientific problems in many areas such as malaria studies, economics studies, and randomized trials with noncompliance. I will propose an EM-like algorithm to numerically calculate the maximum smoothed likelihood estimates of the unknown parameters and functions. I will investigate the convergence of the algorithm and the asymptotic properties of the maximum smoothed likelihood estimators. This research program will not only solve the problem of interest but is also expected to promote the application of these methods in other research areas. The projects in this proposal aim to address challenging problems in various research areas. They will use advanced statistical tools such as mixture models, the empirical likelihood, and the smoothed likelihood to tackle difficult scientific problems. This research program will in turn benefit theoretical and methodological research into these methods. The outcome of both projects will be applicable to many areas related to NSERC, such as those outlined above.