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# Atul Mallik

Aug

15

2013

**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

**Thursday, August 15 2013 at 11:00 a.m.**

Date/Time:

Date/Time:

**438 West Hall**

Location:

Location:

**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.