Abstract

The purpose of this dissertation is to study the first order autoregressive model in the spatial context with specific error structures. We begin by supposing that the error structure has a long memory in both the i and the j components. Whenever the model parameters alpha and beta equal one, the limiting distribution of the sequence of normalized Fourier coefficients of the spatial process is shown to be a function of a two parameter fractional Brownian sheet. This result is used to find the limiting distribution of the periodogram ordinate of the spatial process under the null hypothesis that alpha equals one and beta equals one. We then give the limiting distribution of the normalized Fourier coefficients of the spatial process for both a moving average and autoregressive error structure. Two cases of autoregressive errors are considered. The first error model is autoregressive in one component and the second is autoregressive in both components. We show that the normalizing factor needed to ensure convergence in distribution of the sequence of Fourier coefficients is different in the moving average case, and the two autoregressive cases. In other words, the normalizing factor differs in each of these three cases. Finally, a specific case of the functional central limit theorem in the spatial setting is stated and proved. The assumptions made here are placed on the autocovariance functions. We then discuss some specific examples and provide a test statistics based on the periodogram ordinate.

Notes

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Graduation Date

2019

Semester

Fall

Advisor

Richardson, Gary

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0007772

URL

http://purl.fcla.edu/fcla/etd/CFE0007772

Language

English

Release Date

December 2020

Length of Campus-only Access

1 year

Access Status

Doctoral Dissertation (Open Access)

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Included in

Mathematics Commons

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