Empirical likelihood with infinitely many constraints

经验似然法是对具有有限多参数约束的模型进行推理的一种有效方法。这种非参数最大化方法最初是由Owen提出的，目的是为了构造底层参数的置信区域。同时，经验似然法也能有效地进行估计和检验。需要对该方法进行一般化，允许半参数约束和允许无限多个（参数或半参数）约束。研究者将经验似然的范围扩展到这两个方向。该研究提出了半参数模型的估计理论，并为有效分析各种具体问题中的数据提供了新的方法。在此过程中，求解了递增维二次型的中心极限定理等独立感兴趣的技术问题。半参数模型广泛应用于许多使用统计学的领域。虽然本研究是理论性的，但它为所有这些领域提供了更有效的推理方法，具有很强的实际影响。例如，时间序列的结果在经济预测和数学金融中有应用；二元模型的结果在精算科学和医学研究中都有应用。在医学研究中，双变量数据自然出现在治疗前和治疗后的测量中。这项研究将为研究生在工业界和学术界的职业生涯提供充足的机会。

A powerful method to do inference in models with finitely many parametric constraints is the empirical likelihood approach.This nonparametric maximization method was originally introduced by Owen in order to construct confidence regions for the underlying parameter.In the mean time the empirical likelihood method has been shown to also result in efficient estimation and testing. Needed are generalizations of this method that allow for semiparametric constraints and allow for infinitely many (parametric or semiparametric) constraints.The investigator extends the scope of the empirical likelihood into these two directions. This research advances the theory of estimation in semiparametric models and provides new methods to efficiently analyze data in a wide array of concrete problems. In the process technical problems of independent interest such a central limit theorems for quadratic forms with increasing dimensions are solved.Semiparametric models are widespread in many fields that use statistics.Although this research is theoretical in nature, it has a strong practical impact by providing more effective inference methods for all those fields.For example, results on time series have applications in economic forecasting and in mathematical finance; results on bivariate models have applications in actuarial sciences and in medical research. In medical research bivariate data naturally arise as pre- and post-treatment measurements.The research will provide ample opportunities to prepare graduate students for careers in both industry and academics.