Distance-based robust inferences and model selection for semiparametric models

我的目标是开发强大的统计推断来解决以下问题。急性白血病可根据患者数千个人类基因表达水平中所含的遗传信息分为髓细胞白血病或淋巴细胞白血病。观察到一些患者的一些基因表达异常高/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。在诸如此类的真实的生活问题中，大多数研究都关注解决方案的效率（尽可能使用数据中包含的最多信息），但低估了偏差的影响，包括经常存在的离群观测和模型错误设定。例如，一些数据被污染或记录不正确，或者为差异表达基因假设的模型不是严格有效的。因此，在实践中强烈需要针对这些偏差的鲁棒推断。这些稳健的推论应该总是在假定模型的邻域内很好地工作，并且不会受到模型错误指定和外围观测的过度影响。我提出的研究完全符合这一目标，并将满足这一需求。* *在该主题的大部分历史中，这些都是参数。然而，在过去的四十年中，半参数模型蓬勃发展，归因于其灵活性和可解释性。著名的半参数模型包括生存分析中的考克斯比例风险模型、经济学中的指数模型等。当存在上述偏差时，经典方法如最大似然估计将偏离真实参数值。因此，本研究的目的是建立和调查的一般形式和特定形式的半参数模型的鲁棒性和有效的推理，并审查所提出的推理方法在各个领域的变化和应用。为此，我建议使用最小（Hellinger）距离的方法，几乎没有尝试半参数模型。** 本文提出的鲁棒推理可用于解决医学、遗传学、生存分析、计量经济学、天体物理学等领域的真实的生活问题。通过开发强大的推理，遗传学家可以轻松处理不寻常的观察结果，并提高白血病患者分类的准确性。利用所提出的鲁棒模型选择过程，医学科学家可以选择影响患者生存时间或疾病进展的正确重要因素，即使数据是嘈杂的（例如，患者在完成问卷时回忆过去不准确）。

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).