Atul Mallik


Aug
15
2013

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  • Speaker: Atul Mallik
  • Host Department: Statistics
  • Date: 08/15/2013
  • Time: 11:00AM

  • Location: 438 West Hall

  • Description:

    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.

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