A New Treatment to Dimension Reduction via Semiparametrics

首席研究员（P.I.）将研究降维问题中的六个主题，开发新的方法并分析其性质和性能。主题一涉及中心空间估计。P.I.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。新观点得到了中心空间中包含所有可能估计量的完备估计量类。此外，它还放宽了现有方法中目前需要的各种条件。最后，说明了各种现有估计器之间的关系，并揭示了使所有这些估计器能够发挥作用的潜在原因。主题二涉及中心平均空间估计。P.I.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种常见条件：线性条件和常方差条件。调查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，因为在这些条件放松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. 如果满足并使用了这些条件，那么与不使用这些条件的相同估计器相比，经典估计器将具有膨胀的方差。这项研究将促使人们重新评估这些流行的条件，并转移目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方法，并提出将空间估计问题转化为便于统计推断的等效参数估计问题。该方法还将建立中心空间估计的半参数效率界，并推导出最优有效估计量。她将进一步提供根-$n$收敛速度和效率的理论证明。主题五涉及中心平均空间的推理和有效估计。P.I.将建立与主题四的平行结果，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。主题六是关于确定约简空间的维度。私家侦探将提出三种适合不同评估程序的方法，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，并解决该领域一些最基本和最突出的问题。随着现代科学技术的发展，数据的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）的数据集很容易包含大量变量，这些变量的降维是不可避免的。这些新方法将在这些领域引起广泛的兴趣并具有重要的应用价值。它们还将激发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.