Statistical inference based on complex survey designs using rank information and order statistics

在许多实验中，实现统计上合理的采样设计的一个重要因素是从采样单元进行测量的成本。有时，研究人员可以获得一些辅助信息，或早期调查的结果，专家意见知识，或来自人口的廉价而有用的测量结果，这些测量结果可以在对样本进行最终测量之前轻松地对样本进行排序。例如，考虑在人群中估计骨矿物质密度的问题。这种研究的对象是丰富的，但通过双X射线吸收测量法对选定的对象进行骨密度测量是昂贵的。因此，重要的是尽量减少这种研究所需的测量次数，而不减少获得的有关人群骨矿物质密度构成的可靠信息量。一个成功的策略是使用专家意见的知识，大量的抽样单位，以确定更有代表性的样本，从人口的昂贵的骨密度测量应收集通过双X射线吸收测量。基于秩的抽样设计提供了一系列技术，以获得和分析这些昂贵的测量与廉价的信息的帮助。 这些设计已在渔业、农业、环境和生态研究中得到应用，如辐射（土壤污染和疾病群）或污染（水污染和作物根部疾病），以及医学研究、机器学习和图像处理。 本研究建议的重点是开发新的方法参数和非参数推断的基础上复杂的调查设计，使用秩信息和顺序统计在有限和无限的人口。数学分析将提供洞察推理的过程中，基于排名的数据，并建议新的方法，有效的推理与这样的样本。主要目标是回答广泛的研究问题，涉及基于等级的数据从贝叶斯和频率论的角度。我还将研究考虑排名者质量、排名者数量、样本大小和排名类数量的优化机会。 这项建议所带来的研究进展将有许多重要的应用。例如，加拿大统计局在设计基于等级信息的调查时可能会对结果感兴趣;加拿大卫生部在水质研究和环境风险监测方面可能会对结果感兴趣;加拿大渔业和海洋部在制定监测渔业相关活动的更好方法方面可能会对结果感兴趣。这一建议不仅将补充和扩展现有的基于等级的抽样设计理论，而且有可能解决在其他情况下的方法问题，其中观察涉及审查和/或长度偏差的数据。

In many experiments, an important factor in implementing a statistically sound sampling design is the cost of taking measurements from sampling units. Sometimes researchers have access to some auxiliary information, or results of earlier surveys, expert-opinion knowledge, or inexpensive and useful measurements from the population that can be used to easily rank sampling units prior to taking final measurements on them. For example, consider the problem of estimation of bone mineral density in a human population. Subjects for such a study are plentiful, but measurement of bone mineral density via dual x-ray absorptiometry on the selected subjects is expensive. Thus, it is important to minimize the number of measurements required for such a study without reducing the amount of reliable information obtained about the bone mineral density makeup of the population. One successful strategy is to use expert-opinion knowledge on a large number of sampling units to identify more representative samples from the population from which the expensive bone mineral density measurement should be collected via dual x-ray absorptiometry. Rank-based sampling  designs provide a collection of techniques to obtain and analyze these expensive measurements with the help of inexpensive information.  These designs have found applications in fisheries, agricultural, environmental and ecological studies such as radiation (soil contamination and disease clusters) or pollution (water contamination and root disease of crops)  as well as in medical research, machine learning and image processing. This research proposal focuses on developing new methodologies for parametric and nonparametric inference based on complex survey designs using rank information and order statistics in finite and infinite populations. The mathematical analysis will provide insight into the process of inference from rank-based data and suggest  new methodologies for efficient inference with such samples. The main objective is to answer broad research problems involving rank-based data from both the Bayesian and frequentist perspectives. I will also study the optimization opportunities that take into account quality of rankers, number of rankers, sample size and number of ranking classes. The research developments coming from this proposal will have many important applications. For example, the results could be of interest to Statistics Canada when designing surveys based on rank information; to Health Canada for water quality studies and environmental risk monitoring;  as well as to Fisheries and Oceans Canada for developing better methods for monitoring fisheries-related activities. The proposal would not only complement and extend the existing theory on rank-based sampling designs, but also has the potential to address methodological problems in other settings where observations involve censored and/or length-biased data.