Robust Inferences Based on Minimum Distance for Semiparametric Models

统计推断必然基于数据的统计模型。在该主题的大部分历史中，这些都是参数化的：产生数据的机制可以通过指定几个真实的参数来识别。在过去的三十年中，非参数和半参数模型蓬勃发展。非参数模型使数据的潜在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，许多学者都考虑了有效估计和自适应估计，但对稳健性的研究较少。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，数据可能会有噪声并受到离群值的影响。当存在离群值时，诸如最大似然估计的经典方法将偏离真实参数值。因此，有一个强大的统计推断，总是工作在假定的模型的邻域内，而不是不适当的影响，由模型误指定和离群观测的强烈需求。沿着这个方向，本研究的目的是发展半参数模型的鲁棒推理。为了这个目的，我建议使用最小距离（MD），特别是最小海林格距离（MHD），半参数模型几乎没有尝试的方法。与参数模型相比，半参数模型中额外的未知非参数分量给基于MD的推断的构造和相应的渐近性质的研究增加了额外的复杂性和难度.本研究主要包括四个部分：A.一般半参数模型MD估计的稳健性和有效性研究; B。半参数回归模型基于MD的稳健有效估计的构造;半参数混合模型基于MD的稳健有效估计的构造;基于MD的鲁棒推理在遗传研究中的应用。这四个部分包括理论的基本发展，特定模型的具体实现，以及对现实问题的实际应用。

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.