Parametric Embedding of Non-parametric Problems

排序数据的统计推断本质上是非参数的。非参数统计的优势在于，它可以在对底层分布进行很少假设的情况下产生有效的过程。另一方面，非参数统计通常不依赖于似然函数的使用，这限制了参数统计中常见的通常推理方法的发展。经验似然方法在这个方向上取得了重大进展。另一个进步包括非参数推理问题的参数嵌入，以弥合参数统计和非参数统计之间的明显差距。我的长期愿景是双重的：第一，通过参数嵌入将各种非参数问题整合到参数统计的主流中；第二，超越并利用嵌入族中的参数方法来获得对各种常见非参数问题的额外见解。利用嵌入方法得到非参数推理的基本结果的关键在于参数族的选择。我的短期目标是首先回顾非参数和半参数推理的重要发展，使用这种参数嵌入方法作为一种通用工具，提供对复杂设置的简化见解，并将最优性论证从参数扩展到非参数和半参数问题。然后，我计划以最近的一些成功为基础，进一步证明似然函数可以为几个非参数问题有效地定义，并获得新的最优性结果。嵌入方法是基于早先的Neyman结果的改编，其中为拟合优度问题定义了一个光滑的替代分布。结合使用Rao分数检验，这导致所考虑的分数函数的渐近分布。这样，就有可能得到弗里德曼的随机块设计检验，并证明其局部最优性。在本提案中，我计划将嵌入方法扩展到涉及左截断和右删节（LTRC）数据的更复杂的设置。同样，我计划进一步利用惩罚似然，以减少所考虑的分数函数的维度。另一个目标是考虑一些现实世界的应用，将这一理论与实践联系起来。就影响而言，这种统一的非参数统计方法是新颖的，并且基于我最近的一些结果。它将为在各种情况下进一步发展方法提供额外的工具。***本提案将培训8名HQP，他们将发展数据统计分析的理论和实践技能，为他们在学术界、政府或私营企业的职业生涯做好准备。

Statistical inference with ranked data is inherently non-parametric. The strength of non-parametric statistics is that it can lead to efficient procedures while making few assumptions on the underlying distributions. On the other hand, non-parametric statistics does not generally rely on the use of a likelihood function, which limits the development of the usual inferential methods common in parametric statistics. Empirical likelihood methods have represented a major advance in this direction. Another advance consists of a parametric embedding of non-parametric inference problems to bridge the apparent gap between parametric and non-parametric statistics. My long term vision is two-fold: first, to integrate various non-parametric problems into the main stream of parametric statistics via a parametric embedding and second, to go beyond and make use of parametric methods in the embedded family to gain additional insight into various common non-parametric problems. The key to deriving fundamental results on non-parametric inference with the embedding approach lies in the appropriate choice of the parametric family. My short-term goals are to first revisit important developments in non-parametric and semi parametric inferences using this parametric embedding approach as a versatile tool that provides simplifying insights into complicated settings and extends optimality arguments from parametric to non-parametric and semi parametric problems. Then, I plan to build on some recent successes and further demonstrate that a likelihood function can be fruitfully defined for several non-parametric problems and to obtain new optimality results. The embedding approach is based on an adaptation of an earlier result of Neyman in which a smooth alternative distribution was defined for a goodness of fit problem. Combined with the use of the Rao score test, this leads to the asymptotic distribution of the score functions considered. In this way, it is possible for example, to obtain Friedman's test for randomized block design and to demonstrate its local optimality properties. In this proposal, I plan to extend the embedding approach to the more complicated setting involving left truncated and right censored (LTRC) data. As well, I plan to further exploit the use of penalized likelihood in order to reduce the dimension of the score functions considered. Another goal is to consider some real-world applications to relate this theory to practice. In terms of impact, this unified approach to non-parametric statistics is novel and is based on some of my recent results. It will provide additional tools for further development of methodology in various settings.***This proposal will train 8 HQP, who will develop skills in theoretical and practical statistical analysis of data, preparing them for careers as the case may be, in academia, government or private enterprise.