Topics on Threshold Estimation, Multistage Methods and Random Fields.

Abstract

We consider the problem of identifying the threshold at which a one-dimensional regression function leaves its baseline value. This is motivated by applications from dose-response studies and environmental statistics. We develop a novel approach that relies on the dichotomous behavior of p-value statistics around this threshold. We study the asymptotic behavior of our estimate in two different sampling settings for constructing confidence intervals. The multi-dimensional version of the threshold estimation problem has connections to fMRI studies, edge detection and image processing. Here, interest centers on estimating a region (equivalently, its complement) where a function is at its baseline level. In certain applications, this set corresponds to the background of an image; hence, identifying this region from noisy observations is equivalent to reconstructing the image. We study the computational and theoretical aspects of an extension of the p-value procedure to this setting, primarily under a convex shape-constraint in two dimensions, and explore its applicability to other situations as well. Multistage procedures, obtained by splitting the sampling budget across stages, and designing the sampling at a particular stage based on information obtained from previous stages, are often advantageous as they typically accelerate the rate of convergence of the estimates, relative to one-stage procedures. The step-by-step process, though, induces dependence across stages and complicates the analysis in such problems. We develop a generic framework for M-estimation in a multistage setting and apply empirical process techniques to describe the asymptotic behavior of the resulting M-estimates. Applications to change-point estimation, inverse isotonic regression and mode estimation are provided. In a departure from the more statistical components of the dissertation, we consider a central limit question for linear random fields. Random fields -- real valued stochastic processes indexed by a multi-dimensional set -- arise naturally in spatial data analysis and thus, have received considerable interest. We prove a Central Limit Theorem (CLT) for linear random fields that allows sums to be taken over sets as general as the disjoint union of rectangles. A simple version of our result provides a complete analogue of a CLT for linear processes with no extra assumptions.