Mathematical Sciences: Biased Sampling, Bump Hunting and Confidence

提案：DMS 9504515主要研究者：Jiayang Sun和Michael Woodroofe机构：密歇根大学标题： 有偏抽样、颠簸搜寻和置信度 摘要： 这项研究将包括制定置信区间 为 预期响应，被视为 设计功能 变量 这将通过结合更高的 阶渐近分析、数值计算和仿真。 渐近分析需要找到近似 随机场最大值的分布，甚至渐近 公式需要一些数值计算。 调查人员将 研究了有偏抽样模型的极大似然估计， 包括一些模型， 受试者自选， 单峰性 与多模态的对比。 他们 将寻求高效 算法、一致性条件和渐近分布 用于估计 误差和 测试 统计 更高 秩序 参数模型置信水平的近似 这 展品依赖性是研究的另一个主要目标。 等 模型包括 马尔可夫链，半马尔可夫 过程和时间 序列分析中的序列模型 一个有希望的方法是 在有符号的根变换上使用Stein's Identity。 此外，本发明还提供了一种方法， 的 调查人员 将 研究 非参数估计 为 半马尔可夫过程和期望 样本量和 操作 时间序列模型的特征。 确定可以附加到 估计或预测是研究的主要目的。 做 这 调查人员会把 概率界 不可观测的估计误差。 他们会把注意力集中在 这两个新的背景都涉及测量， 它们是独立的，但却以一种复杂的方式相互关联。 为 例如， 在空气研究中， 污染， 附近的测量 位置可能是相关的。 该研究还包括研究 从观察性研究中推断， 或排除受试者不完全在控制之下， 实验者 在这种情况下，列入可能取决于其他因素， 比如我们感兴趣的变量 例如，在动物研究中， 更容易找到一个大群的动物比一个 小的。 的 发展 修正原始数据的方法， 估计是研究的一个主要目标。

Proposal: DMS 9504515 PI(s): Jiayang Sun and Michael Woodroofe Institution: Univ. of Michigan Title: Biased Sampling, Bump Hunting, and Confidence Abstract: The research will include the development of confidence bands for the expected response, viewed as a function of the design variables. This will be accomplished through combination of higher order asymptotic analysis, numerical calculation, and simulation. The asymptotic analysis requires finding approximations to the distributions of maxima of random fields, and even the asymptotic formulas require some numerical calculation. The investigators will also study maximum likelihood estimation for biased sampling models, including models in which subjects self-select, and tests for unimodality versus multimodality. They will seek efficient algorithms, conditions for consistency, and asymptotic distributions for estimation error and test statistics. Higher order approximations to confidence levels for parametric models which exhibit dependence are another major objective of the research. Such models include Markov chains, semi-Markov processes, and time sequential models in sequential analysis. A promising approach is to use Stein's Identity on the signed root transformation. In addition, the investigators will study non-parametric estimation for semi-Markov processes and expected sample sizes and operating characteristics for time sequential models. Determining the amount of confidence that may be attached to an estimate or projection is a primary objective of the research. To do this the investigators will place probabilistic bounds on the unobservable estimation error. They will focus their attention on two novel contexts both of which involve measurements which are not independent but, rather, are related in a complicated way. For example, in studies of air pollution, measurements at nearby locations may be related. The research also includes the study of inference from observational studies, studies in which the inclusion or exclusion of subjects is not entirely under the control of the experimenter. In such cases inclusion may depend on other factors, like the variables of interest. For example, in animal studies it is easier to find a large group of animals than a small one. The development of methods for correcting the raw data to form valid estimates is a major objective of the research.