The project deals with boundary effects in a class of parametric and nonparametric models. Three main directions are pursued. The first involves the development of methodological and inferential techniques for estimating boundaries in the covariate space (in regression models) using both likelihood based methods and least squares type criteria and are motivated primarily by (a) the need to build stochastic models in response--covariate studies and survival analysis that cater to the development of individualized treatment therapies for cancer-afflicted individuals, and, (b) the development of adaptive multistage strategies for estimating discontinuities in regression surfaces. The second direction involves the determination of a region enclosed by a closed surface in Euclidean space where a function assumes an extremal value (a maximum or a minimum). A novel procedure based on P--values obtained from statistical tests conducted at different points in the domain of the function, for the hypothesis that the function assumes its extreme value at those points, is developed to this end. The final direction deals with quantifying the effect of the sampling distribution of covariates on nonparametric MLEs of the regression function, in nonparametric regression under shape--constraints like monotonicity or convexity. The resolution of the grid on which the covariate is supported is seen to be critically related to the asymptotic distributions of these shape-constrained estimators. Here, one encounters boundary effects in the sense that there exist specific resolutions at which the asymptotics transition dramatically from Gaussian to non-Gaussian.
The proposed research program is motivated by compelling problems in several different areas: from clinical trials for cancer patients, epidemiological studies and data from complex `omics' experiments to problems in systems engineering, signal processing and FMRI studies. On the scientific front, the research based on this grant will lead to new design-based adaptive procedures for studying the behavior of engineering systems under different input intensities as well as new designs for clinical trials to identify core factors involved in cancer progression, to the development of indvidualized treatment therapies in cancer research, to a better understanding of brain activation mechanisms via accurate identification of signals obtained from FMRI and to methods for signal processing via sensor networks. The novel methodological procedures ensuing from our research will be disseminated to the relevant scientific communities, both via inter--disciplinary interaction and collaboration, and the development of software in a readily accessible language environment. Finally, on the educational front, the material from the proposed research will provide dissertation topics for graduate students who will also be supported on this grant; the project will therefore play an important role in the training of future statisticians.
The project dealt with identifying boundaries/thresholds in a variety of problems that arise in a range of disciplines: pharmacology, biomedicine, engineering. For example, in pharmacology a question of key interest is at which level a dose starts showing a detectable effect. This is called the minimum effective dose problem; identifying this threshold helps in designing drugs for patients. In signal processing, it is often of interest to determine the location of a target in a field equipped with wireless sensors that pick up information about potential targets. In environmental applications, it is often important to understand the contours of a pollution cloud or an oil spill from sensor based measurements. We developed new techniques for identifying the nature of such thresholds or boundaries and provided reliability guarantees for our estimates of such thresholds. We also studied how available budgetary resources can be effectively utilized to enhance the quality of our estimation procedures. Our methods will be useful in pollution control. drug development, in strengthening security technology and public health. The use of rigorous mathematical techniques leads to an effective calibration of the reliability of our proposed methods. Our results have been disseminated both to the academic community and industry so that they can trigger further work on these important problems and be used by practitioners.
