Some contributions to stochastic geometry

Wen-Chi Tsai, Purdue University

Abstract

In Part I, we consider an interesting problem with many applications. The canonical problem may be stated as follows: in an Euclidean space Rk, n independent observations uniformly distributed in an unknown compact set S are available. We need to estimate the set S. The work is divided into modelling, estimation, asymptotics, accuracy assessment, and computation. We suggest parametric modelling by using Lp balls as models for the set S. Three methods of estimation are investigated: the maximum likelihood, Bayesian procedures, and a composition of these two. The asymptotic theory presented is of two types: strong consistency and weak convergence. The weak convergence results show that the standard central limit theory does not apply, and the asymptotics are of the extreme value type. The results reveal that Bayes procedures do better, even if the maximum likelihood estimates are corrected for bias. Accuracy in estimating the true set S is assessed by the Hausdorff metric and measures of symmetric differences. Rates of convergences of these are also obtained. In Part II, we consider the Hotelling Beach model of spatial competition on the optimum location for the first firm if customers visit the geographically closest firm, and if there are n future competitors. We study this problem from a statistical perspective for both one and two dimensional markets. The minimax and a Bayesian formulation are considered. The first firm assigns a prior distribution F with density f on the location Z of a customer. In the Bayesian formulation, the first firm assumes that the locations of its competitors are i.i.d. with some distribution G having a density g, and are independent of Z. The decisive factors are the number of future competitors n, and whether or not f and g have appropriate symmetry or unimodality. Specifically, when the number of competitors is large, the first firm's optimum location often moves towards the boundary of S. In one dimension, we are able to give an asymptotic representation for the optimum location of the first firm. The results also suggest that for two dimensional markets, the shape of S does not much affect the first firm's optimum action.

Degree

Ph.D.

Advisors

DasGupta, Purdue University.

Subject Area

Statistics

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