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，它可以导致有效的程序，同时对潜在的分布做出很少的假设。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断方法的发展。经验似然方法代表了这一方向的重大进展。另一个进步包括非参数推断问题的参数嵌入，以弥合参数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，超越并利用嵌入式家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。用嵌入方法导出非参数推断的基本结果的关键在于参数族的适当选择。我的短期目标是首先重新审视非参数和半参数推断的重要发展，使用这种参数嵌入方法作为一种多功能工具，可以简化复杂设置的见解，并将最优性参数从参数扩展到非参数和半参数问题。然后，我计划建立在最近的一些成功，并进一步证明，似然函数可以卓有成效地定义为几个非参数问题，并获得新的最优性结果。嵌入方法是基于适应早期的结果内曼，其中一个光滑的替代分布被定义为一个良好的拟合问题。结合使用的Rao评分测试，这导致考虑的评分函数的渐近分布。通过这种方式，例如，可以获得随机区组设计的Friedman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失（LTRC）数据的更复杂的设置。同时，我计划进一步利用惩罚似然的使用，以减少考虑的得分函数的维度。另一个目标是考虑一些现实世界的应用程序，将此理论与实践联系起来。就影响而言，这种非参数统计的统一方法是新颖的，并且基于我最近的一些结果。它将为在各种情况下进一步制定方法提供更多的工具。 该提案将培训8名HQP，他们将发展数据的理论和实践统计分析技能，为学术界，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.