acfb0e6efca20410VgnVCM100000c2b1d38dRCRDapproved/UMICH/stats/Home/News & Events/Dissertations and Oral Preliminary ExaminationsAtul Mallik###@###(Thu, 15 Aug 2013)Atul Mallik###@###(Thu, 15 Aug 2013)438 West HallTopics on threshold estimation, multistage methods and random fieldsstats1376578800000137657880000011:00AM<p><b>Title:</b> Topics on threshold estimation, multistage methods and random fields</p>
<p><b>Co-Chairs:</b> Associate Professor Moulinath Banerjee <br>
Professor Emeritus Michael B. Woodroofe<br>
<br>
<b>Cognate Member:</b> Professor Alexander Barvinok<br>
<b><br>
Members:</b> Professor Robert W. Keener<br>
Professor George Michailidis<br>
<b><br>
Date/Time: </b>Thursday, August 15 2013 at 11:00 a.m.<br>
<b><br>
Location: </b>438 West Hall</p>
<p><b>Abstract:</b></p>
<p>This dissertation addresses problems ranging from threshold estimation in Euclidean spaces to multistage procedures in Mestimation<br>
and central limit theorems for linear random fields.<br>
We, first, consider the problem of identifying the threshold level at which a one-dimensional regression function leaves its<br>
baseline value. This is motivated by applications from dose-response studies and environmental statistics. We develop a novel<br>
approach that relies on the dichotomous behavior of p-value type statistics around this threshold. We study the large sample<br>
behavior of our estimate in two different sampling settings for constructing confidence intervals and also establish certain<br>
adaptive properties of our estimate.<br>
The multi-dimensional version of the threshold estimation problem has connections to fMRI studies, edge detection and image<br>
processing. Here, interest centers on estimating a region (equivalently, its complement) where a function is at its baseline<br>
level. This is the region of no-signal (baseline region), which, in certain applications, corresponds to the background of an<br>
image; hence, identifying this region from noisy observations is equivalent to reconstructing the image. We study the computational<br>
and theoretical aspects of an extension of the p-value procedure to this setting, primarily under a convex shapeconstraint<br>
in two dimensions, and explore its applicability to other situations as well.<br>
Multistage (designed) procedures, obtained by splitting the sampling budget suitably across stages, and designing the sampling<br>
at a particular stage based on information about the parameter obtained from previous stages, are often advantageous<br>
from the perspective of precise inference. We develop a generic framework for M-estimation in a multistage setting and apply<br>
empirical process techniques to develop limit theorems that describe the large sample behavior of the resulting Mestimates.<br>
Applications to change-point estimation, inverse isotonic regression, classification and mode estimation are provided:<br>
it is typically seen that the multistage procedure accentuates the efficiency of the M-estimates by accelerating the rate<br>
of convergence, relative to one-stage procedures. The step-by-step process induces dependence across stages and complicates<br>
the analysis in such problems, as careful conditioning arguments need to be employed for an accurate analysis.<br>
Finally, in a departure from the more statistical components of the dissertation, we consider a central limit question for li near<br>
random fields. Random fields -- real valued stochastic processes indexed by a multi-dimensional set -- arise naturally in spatial<br>
data analysis and image detection. Limit theorems for random fields have, therefore, received considerable interest. We<br>
prove a Central Limit Theorem (CLT) for linear random fields that allows sums to be taken over sets as general as the disjoint<br>
union of rectangles. A simple version of our result provides a complete analogue of a CLT for linear processes with a<br>
lot of uniformity, at the expense of no extra assumptions.<br>
The oral defense will concentrate on multistage estimation procedures and the problem of estimating convex baseline regions.</p>Nmakingjjsantos13753778462007cfb0e6efca20410VgnVCM100000c2b1d38d____once11112newnewEvent Flyer/UMICH/stats/Home/News & Events/Dissertations and Oral Preliminary Examinations/AtulMallikDefenseFlyer.pdfAtul Mallik