Bootstrap and Threshold Models in Non-standard Problems

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This proposed research deals with methodological and inferential strategies in some non-standard problems that arise in certain non-parametric scenarios. The "non-standard" problems include situations exhibiting non-standard asymptotics -- where estimators converge at rates different from the usual square-root-n rate and/or have non-normal limit distributions. In this proposal, the investigator studies three core directions of statistical research. These are: (A) (In)-consistency of different resampling methods in "non-standard" problems, (B) Estimation and inference with shape restricted functions (where knowledge on the shape of the function, like monotonicity/convexity, is incorporated in estimation), and (C) Estimation of an appropriate "threshold" in the domain of a function where sharp and potentially substantial changes ("regime changes") occur. There is an inherent lack of "smoothness" in these problems (sometimes called the "sharp-edge effect") that manifests in the non-standard rates of convergence and the non-normal limit distributions. Statistical inference in these "non-standard" problems is difficult as the asymptotic distribution theory is complex (and in some cases unknown) with complicated limit distributions, containing nuisance parameters. Bootstrap methods are a natural alternative and are generally reliable in "regular" square-root-n convergence problems. Although there has been extensive activity in the last two/three decades in understanding the behavior of bootstrap in different "regular" scenarios, there has not been much work in such "non-standard" problems, justifying the research projects undertaken in the proposal. The study of the problems has been greatly stimulated by an astronomy collaboration investigating the dark matter content and distribution in dwarf spheroidal (dSph) galaxies. Recent estimates show that the universe consists of about 96% dark matter and dark energy, though very little is known about them as yet. The dSph galaxies occupy a special position in this study -- they are supposed to be the smallest systems containing dark matter, and hence the study of these galaxies is of considerable importance in understanding the structure of the universe. The proposed research will also have diverse other applications, ranging from disciplines in public health like biomedical studies and epidemiology to aspects of the social sciences, especially economics. This is because "non-standard" problems arise 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), in the investigation of "sensitive" time periods (affecting health) in the early development of infants (epidemiology), and so on. The frontiers of the proposed research can be extended through incorporation in the Ph.D. level curriculum. Such interdisciplinary research will open up avenues of investigation in other realted areas of universal interest.