Mathematical Sciences: Biased Sampling, Bump Hunting and Confidence

建议：DMS 9504515 Pi(S)：孙家阳和Michael Woodroofe机构：大学。密歇根大学的题目：有偏抽样、寻找凹凸和信心摘要：这项研究将包括发展预期反应的信心带，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实现。渐近分析需要找到随机场极大值分布的近似值，即使是渐近公式也需要一些数值计算。研究人员还将研究有偏抽样模型的最大似然估计，包括受试者自行选择的模型，以及单峰与多峰的测试。他们将寻求有效的算法，一致性条件，以及估计误差和测试统计的渐近分布。具有相关性的参数模型的置信度的高阶近似是本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使用Stein的身份进行有符号的根变换。此外，研究人员还将研究半马尔可夫过程的非参数估计以及时序模型的预期样本量和运行特性。确定估计或预测可能附带的置信度是这项研究的主要目标。为了做到这一点，调查人员将对不可观测的估计误差设定概率界限。他们将把注意力集中在两个新的背景上，这两个背景都涉及的测量不是独立的，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，在附近地点的测量可能是相关的。这项研究还包括从观察性研究中推断的研究，在这些研究中，受试者的纳入或排除并不完全受实验者的控制。在这种情况下，纳入可能取决于其他因素，如感兴趣的变量。例如，在动物研究中，发现一大群动物比发现一小群动物更容易。开发修正原始数据以形成有效估计的方法是这项研究的主要目标。

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.