In this dissertation, we utilize the discrete Fourier analysis on axially symmetric data generation and nonparametric estimation. We first represent the axially symmetric process as Fourier series on circles with the Fourier random coefficients expressed as circularlysymmetric complex random vectors. We develop an algorithm to generate the axially symmetric data that follow the given covariance function. Our simulation study demonstrates that our approach performs comparable with the classical approach using the given axially symmetric covariance function directly, while at the same time significantly reducing computational costs. For the second contribution of this dissertation, we apply the discrete Fourier transform to provide the nonparametric estimation on the covariance function of the above circularly-symmetric complex random vectors under gridded data structure. Our results show that these estimates has closely related to the simultaneous diagonalization of circulant matrices. The simulation study shows that our proposed estimates match well with their theoretical counterparts. Finally through the Fourier transform of the original gridded data, the covariance estimator of an axially symmetric process based on the method of moments can be represented as a quadratic form of transformed data that is associated with a rotation matrix.
| Identifer | oai:union.ndltd.org:MSSTATE/oai:scholarsjunction.msstate.edu:td-2518 |
| Date | 11 August 2017 |
| Creators | Wang, Wenshuang |
| Publisher | Scholars Junction |
| Source Sets | Mississippi State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Theses and Dissertations |
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