DMS 9626347 Woodroofe The research concerns biased sampling models and exponential time series. In biased sampling models, the probability of including a subject in the study may depend on variables of interest--for example, the size of the subject. Interest centers on cases in which the inclusion probabilities depend on the variables of interest in a monotone way, but without assuming that the inclusion probabilities are known or known up to a few parameters. The research involves developing tests for the presence of bias, developing estimation procedures when bias is present and studying the properties of both tests and estimators. Likelihood and penalized likelihood are used to develop the tests and estimators, and the properties are studied through a combination of asymptotic analysis and simulation. An exponential time series is a stochastic process whose finite dimensional distributions form exponential families. For such process, the sampling distributions of normalized maximum likelihood estimators are asymptotically normal, under quite general conditions. The research involves developing higher order approximations to the these sampling distributions and using the refined approximations to form corrected confidence sets. Mathematically, the results take the form of very weak expansions in which a Bayesian approach is used to obtain approximations to sampling distributions. Biased samples arise frequently in investigations that involve searching for hidden objects, since the probability of finding an object may depend on its properties. For example, astronomers are more likely to find a large bright galaxy than a small dim one, and geologists are more likely to find a large oil well than a small one. Previous work on such problems has concentrated on the case in which the inclusion probabilities depend on the variables of interest in a known way. The new research involves developing data analyses that are appr opriate when the latter relationship is not known and includes methods for estimating the relationship from observed data. The relationship is important, because the number of objects not found depends on it in crucial way. The research considers exponential time series which include classical time series, like stock prices and weather, but also many others like adaptively designed experiments (experiments that design themselves) and sequential clinical trials.
