Title: Topics on Random Fields, Threshold Estimation and Mixture Models
Advisors: Associate Professor Moulinath Banerjee, Professor Michael B. Woodroofe
Committee Members: Professor George Michailidis, Assistant Professor Bodhisattva Sen (Columbia Univ)
Abstract: Random fields have attracted a lot of attention especially in modeling spatially correlated data. Limit theorems for random fields have also motivated a number of papers. We provide sufficient conditions for Central Limit Theorem to hold for linear random fields when the sums are taken over sets of arbitrary shape. Stronger results are obtained for the special cases, namely, rectangles and union of rectangles. The next problem we consider is the estimation of baseline-value regions of a function. In several applications, the data follows a regression model where the regression function is fixed at its baseline value for covariate levels in a certain region and stays above that value otherwise. This is relevant to a broad range of applications. We provide partial results on consistency, rate of convergence and the asymptotic distribution for the proposed estimate in the one dimensional case and discuss directions for future work. Another problem of our interest involves non-parametric maximum likelihood estimation (NPMLE) in mixture models. We mainly discuss two applications: one involving estimation of a criterion for control while classifying bank accounts as sensitive to frauds based on observed number of transactions under certain assumptions and the other involving estimation of distribution of velocity of stars in an isotropic galaxy. The general theory of estimating NPMLE in a mixture model setup is relevant here and has been discussed along with some related problems which will be pursued in future.