Distance-based robust inferences and model selection for semiparametric models

我的目标是开发强有力的统计推断来解决以下问题。根据数千个人类基因表达水平中包含的患者遗传信息，急性白血病可分为髓系白血病和淋巴细胞性白血病。观察到一些患者的某些基因表达异常高/低。这些极端观测既不能因为数据量大而手动删除，也不能因为它们可能对分类结果产生的未知影响而随意删除。在现实生活中，大多数研究关注的是解的效率(尽可能多地利用数据中包含的信息)，而低估了偏差的影响，包括异常观测和模型错误描述，这些偏差经常存在。例如，一些数据被污染或记录错误，或者假设差异表达基因的模型不严格有效。因此，在实践中迫切需要针对这些偏差做出强有力的推断。这些稳健的推论应该总是在假定的模型的邻域内很好地工作，并且不会受到模型错误指定和离奇观测的过度影响。我提出的研究完全符合这一目标，并将恰好满足这一需求。统计推断必然是以统计模型为基础的。在这一主题的大部分历史中，这些都是参数。然而，在过去的四十年里，半参数模型因其灵活性和可解释性而蓬勃发展。著名的半参数模型包括生存分析中的Cox比例风险模型、经济学中的指数模型等。当存在上述偏差时，最大似然估计等经典方法会偏离真实参数值。因此，本研究旨在构建和研究一般形式和特殊形式的半参数模型的稳健和有效的推理，并仔细检查所提出的推理方法在不同领域的变化和应用。为此，我建议使用最小(Hellinger)距离方法，这种方法在半参数模型中几乎没有尝试过。这项建议中提出的稳健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现实生活问题。以下是两个示例。有了这些强大的推论，遗传学家可以轻松地处理不寻常的观察结果，并比常用的方法提高白血病患者分类的准确性。通过建议的稳健模型选择程序，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，即使数据有噪声(例如，患者在完成调查问卷时不准确地回忆过去)。

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).