A New Treatment to Dimension Reduction via Semiparametrics

首席调查员(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。主题一涉及中心空间估计。P.I.将启动一种与当前文献完全不同的方法，使用半参数治疗。新的视点导致了中心空间的一类完整的估计量，其中包含所有可能的估计量。此外，它还放宽了现有方法目前所要求的各种条件。最后，说明了现有的各种估计量之间的关系，并揭示了这些估计量得以发挥作用的根本原因。主题二涉及中心均值空间估计。P.I.将建立与主题1类似的结果，使用相同的一般概念，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。P.I.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，一致估计仍然可以进行。它们在两个方面都是有害的。1.如果不满足这些条件，但假设是错误的，则经典估计是有偏的。2.如果满足并使用这些条件，与不使用这些条件的相同估计相比，经典估计将具有夸大的方差。这项研究将促使对这些流行条件的重新评估，并引导当前进一步开发这些条件的研究趋势。主题四涉及中心空间的统计推断和有效估计。P.I.将设计一种方便的参数化法，并建议将空间估计问题转化为便于统计推断的等价参数估计问题。P.I.还将建立中心空间估计的半参数效率界，并将得到最优有效估计器。她将进一步提供根-$n$收敛速度和效率的理论证明。主题五涉及中心均值空间的推断和有效估计。P.I.将建立与主题四类似的结果，说明这两个问题之间的差异，并突出它们截然不同的分析结果。主题六涉及确定缩减空间的维度。P.I.将针对不同的估计过程提出三种方法，并检查它们的有效性。该提案中的一系列项目将为降维问题提供新的线索，并解决该领域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、越来越复杂已成为一种普遍现象。来自医学、遗传学、环境科学、包括互联网行为研究在内的社会行为科学的数据集容易包含大量的变量，降维是不可避免的。新的方法将引起广泛的兴趣，并在这些领域有重要的应用。它们还将促进统计科学本身在相关半参数问题和计算方法方面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.