Mathematical Sciences: Biased Sampling, Bump Hunting and Confidence

提案：DMS 9504515 PI(s): jiayyang Sun and Michael Woodroofe Institution: university of Michigan题目：有偏抽样、Bump Hunting和置信度摘要：研究将包括期望响应的置信度的发展，将其视为设计变量的函数。这将通过高阶渐近分析、数值计算和模拟相结合来完成。渐近分析需要找到随机场最大值分布的近似值，甚至渐近公式也需要一些数值计算。研究者还将研究有偏抽样模型的最大似然估计，包括受试者自我选择的模型，以及单模态与多模态的检验。他们将寻求有效的算法，一致性的条件，估计误差和检验统计量的渐近分布。对表现出依赖性的参数模型的置信水平的高阶近似是研究的另一个主要目标。这些模型包括序列分析中的马尔可夫链、半马尔可夫过程和时间序列模型。一种很有前途的方法是在有符号根变换上使用斯坦恒等式。此外，研究人员将研究半马尔可夫过程的非参数估计以及时间序列模型的预期样本量和运行特性。确定估计或预测可能附加的置信度是研究的主要目标。为了做到这一点，研究人员将在不可观察的估计误差上设置概率界限。他们将把注意力集中在两种新的环境上，这两种环境都涉及到不是独立的测量，而是以一种复杂的方式联系在一起的。例如，在空气污染的研究中，附近地点的测量可能是相关的。该研究还包括对观察性研究的推断研究，在这些研究中，受试者的纳入或排除并不完全在实验者的控制之下。在这种情况下，包括可能取决于其他因素，如感兴趣的变量。例如，在动物研究中，找到一大群动物比找到一小群动物更容易。开发校正原始数据以形成有效估计的方法是本研究的主要目标。

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.