A New Treatment to Dimension Reduction via Semiparametrics

主要研究者（P.I.）将研究降维问题中的六个主题，开发新的方法并分析它们的性质和性能。主题一是中心空间估计。私家侦探将启动一个完全不同的方法，从目前的文献，使用半参数处理。新的观点的结果在一个完整的一类估计的中心空间，其中包含所有可能的估计。此外，它放宽了现有方法中目前所需的各种条件。最后，它说明了各种现有的估计之间的关系，并揭示了潜在的原因，使所有这些估计功能。主题二：中心平均空间估计。私家侦探我将建立与主题一平行的结果，使用相同的一般思想，但通过不同的分析推导。主题三涉及降维中的两个常见条件：线性条件和常方差条件。私家侦探将揭示一个惊人的发现，这些条件不仅是多余的，而且是有害的。它们是多余的，因为在这些条件放宽的情况下仍然可以进行一致的估计。它们在两个方面是有害的。1.如果不满足这些条件，而是错误地假设，则经典估计量是有偏的。2.如果满足这些条件并使用这些条件，则与不使用这些条件的相同估计量相比，经典估计量将具有膨胀的方差。这项研究将促使重新评估这些流行的条件，并转移目前的研究趋势，进一步利用这些条件。主题四涉及统计推断和中心空间的有效估计。私家侦探将设计一个方便的参数化，并建议将空间估计问题转化为一个等价的参数估计问题，便于统计推断。私家侦探还将建立中心空间估计的半参数有效界，并将导出最佳有效估计量。她将进一步提供理论证明的根$n$收敛速度和效率。主题五是中心均值空间的推断和有效估计。私家侦探将建立与主题四平行的结果，说明两个问题之间的差异，并强调它们截然不同的分析结果。题目六涉及确定约化空间的维数。私家侦探将针对不同的估计程序提出三种方法，并检验它们的实用性。本提案中的一系列项目将为降维问题提供新的思路，并解决这一领域中一些最基本和最突出的问题。伴随着现代科学技术的发展，数据越来越丰富和复杂已经成为一种普遍现象。来自医学、遗传学、环境科学、社会行为科学包括网络行为研究的数据集容易包含大量的变量，降维是不可避免的。新的方法将在这些领域引起广泛的兴趣并有重要的应用。这些结果也将促进统计科学中相关半参数问题和计算方法的进一步研究和发展。

The principal investigator (P.I.) will study six topics in dimension reduction problems, develop new methodologies and analyze their properties and performances. Topic One concerns central space estimation. The P.I. will initiate a totally different approach from the current literature, using a semiparametric treatment. The new view point results in a complete class of estimators for the central space which contains all possible estimators. In addition, it relaxes various conditions currently required in the existing methods. Finally, it illustrates the relations between various existing estimators and reveals the underlying reason which enables all these estimators to function. Topic Two concerns central mean space estimation. The P.I. will establish parallel results to Topic One, using the same general idea but via different analytic derivation. Topic Three concerns two common conditions in dimension reduction: the linearity condition and the constant variance condition. The P.I. will reveal an astonishing discovery that these conditions are not only redundant, but also detrimental. They are redundant in that consistent estimation can still be carried out of these conditions are relaxed. They are detrimental in two ways. 1. If these conditions are not met, but are falsely assumed, the classical estimators are biased. 2. If these conditions are met and are used, the classic estimators will have inflated variances compared to the same estimator without using these conditions. This research will prompt re-evaluation of these popular conditions and divert the current research trend of further exploiting these conditions. Topic Four concerns statistical inference and efficient estimation of the central space. The P.I. will devise a convenient parameterization and proposeto convert the space estimation problem to an equivalent parameter estimation problem which facilitates statistical inference. The P.I. will also establish the semiparametric efficiency bound for the central space estimation and will derive the optimal efficient estimator. She will further provide theoretical proof of the root-$n$ convergence rate and the efficiency. Topic Five concerns inference and efficient estimation for the central mean space. The P.I. will establish parallel results to Topic Four, illustrate the difference between the two problems and highlight their drastically different analytic results. Topic Six concerns deciding the dimension of the reduced space. The P.I. will propose three methods tailored to different estimation procedures and examine their usefulness.The series of projects in this proposal will shed new light on the dimension reduction problems and resolve some of the most fundamental and outstanding issues in this field. Accompanying the modern development of sciences and technologies, richer and more complex data have become a common phenomenon. Data sets from medical science, genetics, environmental science, social behavioral science including internet behavior studies easily contain a large amount of variables that dimension reduction is unavoidable. The new methodologies will generate wide interest and have important application in these fields. They will also provoke further studies and development in related semiparametric problems and computing methods in statistical sciences itself.