The primary objectives of the proposal are to study biased sampling models and tests of significance in projection pursuit. For biased sampling models, penalized maximum likelihood estimators will be sought and their properties analyzed through a combination of simulation and asymptotic analysis. Techniques from isotonic estimation are expected to be useful and new techniques will be developed as well. Significance tests for projection pursuit will be studied through simulation and asymptotic analysis, probably drawing on recent work on the maxima of Gaussian random fields. It is conjectured that coupling methods may yield a proof of the Markov Renewal Theorem. Observational studies often encounter or utilize purposively unequal probability sampling models (referred to as biased sampling models). A long term goal of this research is the development of methodology for inference when the (unequal probability) selection mechanism can be modeled. A second aspect of this research considers projection pursuit, that is, the search for underlying relationships in large complex data sets with many cases and with many variables. In particular, criteria for determining when relationships have been recognized accurately will be sought.
