Atul Mallik

Topics on threshold estimation, multistage methods and random fields

Title: Topics on threshold estimation, multistage methods and random fields

Co-Chairs: Associate Professor Moulinath Banerjee 
Professor Emeritus Michael B. Woodroofe

Cognate Member: Professor Alexander Barvinok

Members: Professor Robert W. Keener
Professor George Michailidis

Date/Time: Thursday, August 15 2013 at 11:00 a.m.

Location: 438 West Hall

Abstract:

This dissertation addresses problems ranging from threshold estimation in Euclidean spaces to multistage procedures in Mestimation
and central limit theorems for linear random fields.
We, first, consider the problem of identifying the threshold level 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 type statistics around this threshold. We study the large sample
behavior of our estimate in two different sampling settings for constructing confidence intervals and also establish certain
adaptive properties of our estimate.
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. This is the region of no-signal (baseline region), which, in certain applications, 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 shapeconstraint
in two dimensions, and explore its applicability to other situations as well.
Multistage (designed) procedures, obtained by splitting the sampling budget suitably across stages, and designing the sampling
at a particular stage based on information about the parameter obtained from previous stages, are often advantageous
from the perspective of precise inference. We develop a generic framework for M-estimation in a multistage setting and apply
empirical process techniques to develop limit theorems that describe the large sample behavior of the resulting Mestimates.
Applications to change-point estimation, inverse isotonic regression, classification and mode estimation are provided:
it is typically seen that the multistage procedure accentuates the efficiency of the M-estimates by accelerating the rate
of convergence, relative to one-stage procedures. The step-by-step process induces dependence across stages and complicates
the analysis in such problems, as careful conditioning arguments need to be employed for an accurate analysis.
Finally, in a departure from the more statistical components of the dissertation, we consider a central limit question for li near
random fields. Random fields -- real valued stochastic processes indexed by a multi-dimensional set -- arise naturally in spatial
data analysis and image detection. Limit theorems for random fields have, therefore, 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 a
lot of uniformity, at the expense of no extra assumptions.
The oral defense will concentrate on multistage estimation procedures and the problem of estimating convex baseline regions.