DMS 9626118 Brown This research involves three statistical topics. i) There are several useful formulations involving nonparametric function estimation including nonparmetric regression, nonparametric density estimation and the standard nonparametric signal processing model. Recently the investigator and others have proved the asymptotic equivalence of these models in settings involving observation of one dimensional response variables. The investigators propose to construct a formulation for asymptotic equivalence to enable obtaining effective equivalence results for multidimensional response variables. ii) One standard method for constructing statistical tests in certain complex parametric situations involves conditioning on an ancillary statistic and constructing a test within the (simpler) conditional problem at the desired level of statistical significance. Unfortunately this procedure entails some peculiar statistical decision theoretic consequences. These are investigated more fully and alternatives are proposed to this standard methodology which are nearly as simple to implement but avoid these undesirable consequences. iii) Standard tests for bioequivalence are inherently one - dimensional in that they look at only one performance aspect at a time. This research constructs multidimensional bioequivalence procedures based in part on principles contained in recent research by the investigator and collaborators. Statistical procedures are often classified as single parameter, multiparameter, or nonparametric depending whether the research looks respectively at one performance characteristic, several related ones, or an unstructured continuum of them, such as a response function over time of unspecified shape. i) An important component of the modem biotechnology enterprise involves testing whether newly developed replicas of existing technologies have the same effect as the original. For example, does a (cheaper) generic drug work exactly t he same as the original prescription drug it is intended to replace. Standard statistical tests of "bioequivalence" have been formulated to test this hypothesis based on data gathered about the two technologies. These standard tests are inherently one-dimensional, and can examine only one performance facet at a time. This research involves the construction of general statistical tests of bioequivalence which can simultaneously examine several related performance aspects. ii) Nonparametric function estimation lies at another extreme in statistical methodology. Such techniques involve little prior structure and few apriori assumptions. They are thus extremely valuable in a variety of modern scientific enterprises involving analysis of masses of complex data including research on the environment and on global change. One technical problem impacting this area has been the existence of several apparently different mathematical formulations for such problems. This research involves the creation of a unified theory for such problems which will enable more efficient research and more effective use of these methods. iii) A third aspect of this research involves the notion of asymptotically ancillary statistics. These are a technical device invented in part to simplify the analysis of complex multiparameter models. Modern data collection and computing have vastly increased the importance of analyzing such models, but more familiarity with them reveals certain peculiarities and deficiencies which result from using this technical device. This research proposes alternate technical advice which is nearly as simple to use but avoids these peculiarities.
