Distance-based robust inferences and model selection for semiparametric models

我的目标是开发可靠的统计推断来解决以下问题。根据数千种人类基因表达水平中包含的患者遗传信息，可将急性白血病分为髓系白血病和淋巴母细胞白血病。观察到部分患者的某些基因表达异常高/低。由于数据量大，这些极端观测值既不能手动删除，也不能任意删除，因为它们可能对分类结果产生未知的影响。在诸如此类的现实问题中，大多数研究关注的是解决方案的效率（尽可能多地使用数据中包含的信息），但低估了偏差的影响，包括经常存在的离群观测值和模型错配。例如，一些数据被污染或记录不正确，或者假设的差异表达基因模型并不严格有效。因此，在实践中强烈需要针对这些偏差的稳健推断。这些稳健的推断应该总是在假定模型的邻域内工作得很好，并且不会受到模型错误说明和外围观测值的过度影响。我提出的研究完全符合这个目标，并将完全满足这一需求。

I aim to develop robust statistical inferences to solve problems such as the following. Acute leukemia can be classified as either myeloid or lymphoblastic leukemia based on patients' genetic information contained in the expression levels of thousands of human genes. It is observed that some patients' some genes are expressed unusually high/low. These extreme observations can be removed neither manually due to the large size of data nor arbitrarily due to the unknown influence they may have on classification result. In real life problems such as this, most researches focus on the efficiency (using as most information contained in data as possible) of solutions but underestimate the influence of deviations, including outlying observations and model misspecification, that often exist. For example, some data is contaminated or incorrectly recorded, or the model assumed for differentially expressed genes is not strictly valid. Therefore, robust inferences against these deviations are strongly desired in practice. These robust inferences should always work well within neighborhoods of the putative model and are not unduly influenced by model misspecification and outlying observations. My proposed research aligns perfectly with this target and will fulfill exactly this demand. Statistical inference necessarily is based on statistical model. During most of the history of the subject, these have been parametric. However, during the last forty years semiparametric models have flourished, attributed to its flexibility and interpretability. Well-known semiparametric models include Cox proportional hazard model in survival analysis, index model in economics, among many others. When deviations above mentioned are present, classical methods such as maximum likelihood estimation will be distorted away from true parameter values. Therefore, this proposed research aims to construct and investigate robust and efficient inferences for semiparametric model both of general form and of particular forms and to scrutinize variations and applications of the proposed inference methodologies in various areas. For this purpose, I propose to use minimum (Hellinger) distance approach that has been barely attempted for semiparametric models. The proposed robust inferences in this proposal can be used to solve real life problems in medical sciences, genetic studies, survival analysis, econometrics, astrophysics and so on. The following are two examples. With the developed robust inferences, geneticists can easily handle unusual observations and improve leukemia patient classification accuracy over commonly used methods. With the proposed robust model selection procedure, medical scientists can select the right significant factors influencing a patient's survival time or disease progress even when the data is noisy (e.g. a patient recalls the past inaccurately when finishing a questionnaire).