Shape-constrained estimation and inference: robustness and adaptability

在当今世界，统计分析正以前所未有的方式推动科学和工程领域的探索和发现。具有优势特征的统计方法对于推动这些进步至关重要。在接下来的五年里，我将进行方法、理论和计算方面的研究，重点是形状约束的非参数方法。近年来，在形状约束领域的研究活动有了戏剧性的和令人兴奋的增长，因此，形状约束方法已经成为传统非参数平滑方法的一个有吸引力的和更好的选择。通过利用形状约束非参数技术的固有特征，通过拟议的研究计划产生的新数据分析方法将具有独特的性能特征组合：它们将是鲁棒的，最大限度地减少用户做出模型假设的需要，它们将是自适应的，在某些用户指定的情况下自然地达到与参数方法相似的效率，它们将是全自动的，导致自然的最优和计算效率高的方法。该提案的具体技术目标是基于形状受限非参数领域的当前开放问题，同时解决重要的数据分析需求，包括（但不限于）保险或各种类型的医学研究中产生的多变量和高维数据集。项目由真实数据驱动，结合观察到的数据特征和重要的和以前未考虑的场景的收集方法。由此产生的方法将提高从业者分析、建模和从数据中提取信息的能力，为进一步的科学发现铺平道路。相关的培训计划将培养具有现代计算、统计和数学方法高级专业知识的高素质人才。获得的技能将使这些HQP能够在统计、金融、公共卫生和数据科学等领域追求成功的学术、研究或工业事业。

In today's world, statistical analysis is driving exploration and discovery in science and engineering in unprecedented ways. Statistical methods with advantageous characteristics are crucial to furthering these advances.  I will conduct methodological, theoretical, and computational investigations focusing in the next five years on shape-constrained nonparametric methods. In recent years, there has been a dramatic and exciting increase in research activity in the area of shape-constraints, and as a result, shape-constrained methods have emerged as an appealing and superior alternative to traditional nonparametric smoothing methods.  By capitalizing on inherent features of shape-constrained nonparametric techniques, the new data analysis methods produced through the proposed research program will have a unique combination of performance characteristics: they will be robust, minimizing the need of the user to make model assumptions, they will be adaptive, naturally reaching similar efficiency to parametric approaches in certain user-specified cases, and they will be fully automatic, leading to naturally optimal and computationally efficient approaches. Specific technical objectives of the proposal are based on current open problems in the field of shape-constrained nonparametrics, while at the same time addressing important data analysis needs including (but not limited to) multivariate and high-dimensional data sets arising in insurance or various types of medical studies. Projects are driven by real data, incorporating observed data characteristics and collection methods for important and previously unconsidered scenarios.  The resulting methods will therefore improve the practitioner's ability to analyze, model, and extract information from data, paving the way for further scientific discovery.  The associated training program will produce highly qualified personnel (HQP) with advanced expertise in specialized modern computational, statistical, and mathematical methods.   The skills gained will enable these HQP to pursue successful academic, research, or industrial careers in fields such as statistics, finance, public health, and data science.