Statistical methodology for rank based sampling design and finite mixture models

在许多研究中，研究变量之间的关系以更好地分类和解释数据集中的同质性和异质性。 例如，在渔业调查中，这种分类可以转化为根据长度-频率数据确定鱼的年龄。在内分泌学中，该分类可用于确定患者的甲状腺疾病状态。在神经生物学中，这种分类可以用来理解与记忆生活事件顺序的能力有关的海马神经活动。 分类和聚类分析是统计科学和许多应用的基本要素。混合概率模型是基于模型的分类和聚类的灵活而强大的统计工具。混合模型的能力允许使用基于似然的方法，这些方法优选用于各种统计推断，包括假设检验、分类、聚类和预测密度估计等。混合模型的灵活性允许分析复杂的数据集。这包括以下情况：1）数据结构本身复杂且具有挑战性，或2）不同的抽样设计更适合目标人群。 随着技术的进步，我们目睹了在产生复杂和大容量数据集的研究和实验方面的巨大进步，例如，在神经成像中，例如功能性磁共振成像（fMRI）数据。我计划研究如何处理高维混合模型，并使用低维模型处理它们。这导致更有效的分类和聚类分析，并使我们能够更好地推断数据的异质性。 在许多统计调查中，研究人员经常面临两个关键挑战：1）测量成本和2）从感兴趣的人群中获得更具代表性的样本。例如，在骨质疏松症研究中，骨疾病的诊断需要通过双X射线吸收测量成像从骨骼部位（如股骨颈和腰椎）测量骨矿物质密度（BMD）。 该过程是昂贵和耗时的;然而，从业者可以获得易于测量的特征，如患者的年龄，体重或基线BMD。基于秩的抽样设计为上述挑战提供了解决方案。这些抽样设计使我们能够使用这些易于测量的信息（例如，年龄）来改进数据收集过程，从而从感兴趣的群体中获得更具代表性的样本（例如，骨病）。 我计划研究如何利用这些信息丰富的基于秩的样本来更好地推断混合模型。

In many research studies, the relationship between variables is investigated to classify and explain better the homogeneities and heterogeneities in datasets. In fishery surveys, for example, this classification can be translated into the age determination of fish from length-frequency data. In Endocrinology, the classification can be used to determine the Thyroid disorder status of patients. In Neurobiology, this classification can be used to understand the hippocampal neural activities that are linked with the ability to remember the order of life events. Classification and cluster analysis are fundamental elements of statistical science and many applications. Mixture probabilistic models are flexible and powerful statistical tools for model-based classification and clustering. The power of mixture models allows the use of likelihood-based methods, which are preferred for various statistical inferences, including hypothesis testing, classification, clustering and predictive density estimation, to name a few. The flexibility of mixture models allows the analysis of complex datasets. This includes the situations where 1) the data structure is itself complex and challenging or 2) a different sampling design is more appropriate for the target population. As technology advances, we are witnessing dramatic progress in research and experiments producing complex and high-volume datasets, e.g., in Neuroimaging such as functional magnetic resonance imaging (fMRI) data. I plan to investigate how we can deal with high dimensional mixture models and handle them with low-dimensional models. This leads to more efficient classification and cluster analysis and enables us to make better inferences about the heterogeneity in data. In many statistical surveys, there are two critical challenges that researchers often face: 1) the cost of taking measurements and 2) obtaining more representative samples from the population of interest. In Osteoporosis research, for example, the diagnosis of bone disorder requires measuring bone mineral density (BMD) through dual X-ray absorptiometry imaging from skeletal sites such as femoral neck and lumber spine. This procedure is costly and time-consuming; however, practitioners have access to easy-to-measure characteristics such as age, weight or baseline BMD from patients. Rank-based sampling designs present a solution to the above challenges. These sampling designs enable us to use this easy-to-measure information (e.g., age) to improve the data collection process leading to more representative samples from the population of interest (e.g., bone disorder). I plan to investigate how we can exploit these informative rank-based samples to make better inferences about the mixture models.