The proposed project deals with methodology and inference strategies for nonparametric and semiparametric problems exhibiting non--standard asymptotics, problems where estimators converge at rates different from the usual rate (square root of the sample size) and/or have non--Gaussian limit distributions. The two core areas of investigation are: (A) Estimation of functions under shape restrictions, and (B) Estimation of an appropriate ``threshold'' in the domain of a function where sharp and potentially substantial changes occur. Part of the emphasis in shape--restricted inference will be on maximum likelihood/least squares based procedures and in particular, the construction of confidence sets for quantities of interest (like a regression function, a hazard function etc.) by inversion of residual sum of squares or likelihood ratio statistics. This is in light of initial exploration for monotone function models where such statistics are seen to exhibit asymptotically pivotal behavior, which facilitates inference, since nuisance parameters need not be estimated from the data. Shape constraints of interest are monotonicity, unimodality and convexity/concavity, and often a combination of such features. Smoothing under shape constraints will also be investigated in some of these problems, since smoothing typically yields faster rates of convergence and provides automatic estimation of derivatives of functions, which are often of interest, for example in economics. On the threshold estimation front, two main areas will be explored. The first is change--point estimation, where there is a jump in the value of the function of interest or in the value of the derivative, with the focus being on design issues: how to design the sampling mechanism, given a fixed budget (of points that can be sampled) so as to entail precise detection of the jump. The second area concerns studying appropriate notions of a threshold for smooth functions that show rapid change over a short domain. One such notion can be formulated in terms of a function with a finite number of discontinuities that is used simply as an approximation, or a working model for the true function. The discontinuities of the best fitting working model provide a natural description of a threshold. Split--point estimation, as this is known, turns out to be radically different from change point estimation, in light of some initial work and will be investigated in more detail.
The proposed research will have diverse applications, ranging from disciplines in public health like biomedical studies and epidemiology to topics in the social sciences (economics) and the physical sciences (astronomy). Shape restrictions, for example, show up naturally in the analysis of productions of firms/companies (economics), the study of the risk of succumbing to illness or infection with age (biomedical research/public health), and problems associated with the detection of dark matter in galaxies, solutions to which can shed light on the future evolution of the universe (astrophysics). The P.I. is actively involved in collaborations with econometricians, epidemologists and astronomers: the research emanating from this grant will therefore have strong interdisciplinary flavor and will address many real scientific questions of interest. On the threshold estimation front, statistical methods for split point detection are of importance, since split points have been used as a measure of threshold by ecologists in the development of pollution control standards. The problems related to change point detection will be applied to detection of stress thresholds in engineering systems. The statistical methodology to be developed will be circulated to the statistical community in academia and industry through free software packages. On the educational front, some of the research material in this proposal will be incorporated in advanced courses at the graduate level and an interdisciplinary seminar series. Some projects will also serve as dissertation topics for Ph.D. advisees and will therefore play an important role in the training of future statisticians.
