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.
