Robust Inferences Based on Minimum Distance for Semiparametric Models

统计推断必然以数据的统计模型为基础。在受试者的大部分历史中，这些都是参数：生成数据的机制可以通过指定几个真实的参数来识别。在过去的三十年里，非参数和半参数模型蓬勃发展。非参数模型使得数据的底层分布基本上是自由的，而半参数模型是介于参数和非参数之间的模型。对于半参数模型，许多作者考虑了有效的、自适应的估计，但对估计的稳健性关注较少。在大多数情况下，假设统计模型对数据进行建模，该模型只是对现实的近似，数据可能是噪声和受异常值的影响。当存在异常值时，最大似然估计等经典方法会偏离真实参数值。因此，对稳健的统计推断有着强烈的需求，这些推断总是在假定的模型的邻域内工作得很好，并且不会受到模型错误指定和离奇观测的过度影响。沿着这一方向，本研究旨在开发半参数模型的稳健推理。为此，我建议使用最小距离(MD)，特别是最小Hellinger距离(MHD)，这种方法在半参数模型中几乎没有尝试过。与参数模型相比，半参数模型中额外的未知非参数分量增加了基于MD的推理的构造和相应渐近性质的研究的复杂性和难度。本研究包括四个部分：A.一般半参数模型的MD估计的稳健性和有效性研究；B.基于MD的半参数回归模型稳健有效估计的构造；C.基于MD的半参数混合模型稳健有效估计的构造；D.基于MD的稳健推断在遗传学研究中的应用。这四个部分包括理论的基本发展，特定模型的具体实施，以及对现实世界问题的实际应用。

Statistical inference necessarily is based on statistical models for data. During most of the history of the subject, these have been parametric: the mechanism generating the data could be identified by specifying a few real parameters. During the last thirty years nonparametric and semiparametric models have flourished. Nonparametric model leaves the underlying distribution of data essentially free, while semiparametric model is a model between parametric and nonparametric. For semiparametric models, many authors have considered efficient and adaptive estimation while the robustness has been paid little attention. In most situations where a statistical model is assumed to model data, the model is only an approximation to reality and the data could be noisy and subject to outliers. When outliers are present, classical methods such as maximum likelihood estimation will be distorted away from the true parameter values. Therefore, there is a strong demand of robust statistical inferences that always work well within neighborhoods of the putative model and are not unduly influenced by model misspecification and outlying observations. Following this direction, this proposed research aims to develop robust inferences for semiparametric models. For this purpose, I propose to use minimum distance (MD), particularly minimum Hellinger distance (MHD), approach that has been barely attempted for semiparametric models. Compared with parametric models, the additional unknown nonparametric component in semiparametric models adds an extra degree of complexity and difficulty to the construction of MD based inferences and to the study of corresponding asymptotic properties. This proposed research includes four parts: A. Robustness and efficiency study of MD estimation for general semiparametric model; B. Construction of robust efficient estimation based on MD for semiparametric regression models; C. Construction of robust efficient estimation based on MD for semiparametric mixture models; D. Applications of MD based robust inferences in genetic studies. These four parts encompass fundamental development in theorem, concrete implementation for specific models, and practical application to real-world problems.