| In this thesis, tests for periodicity are investigated based on a spectral analysis of a time series. Some important fundamental theories of spectral analysis of stationary and harmonic processes are reviewed. A regression model is developed in the frequency domain based on the Fourier transformation. We present some of the periodogram based tests, which are the global test and three local tests, i.e., the hearing test, local F test and Thomson's multitaper test. We will show that most of the tests can be derived from our regression model with the error term having an approximately diagonal covariance matrix. The distribution of the error term of the spectral regression model is based on the asymptotic distribution of the tapered Fourier transform of the error process. This asymptotic distribution has approximately a diagonal covariance matrix when the sample size is large and the spectral density functions (SDFs) of error processes have small dynamic range. We contrast the F test in the time domain and the local F test in the frequency domain as well as the global and local spectral-based tests. The global test uses the spectral estimates at all sample points, whereas a local test uses a subset of the sample points available. Standard global tests for periodicity are often based on the assumption of a Gaussian IID error process. Using a smoothing spline approach, we extend the global test to the non-IID case. We compare this approach to a number of local tests for periodicity such as the local F test, the test commonly used in hearing sciences, and Thomson's multitaper F test. Using regression-based F tests, we demonstrate that asymptotic size and power calculations can be made for some of these tests. We compare the size and power at finite sample sizes, under a number of different experimental conditions.; According to the exploratory data analysis, we applied the local F test to hearing data, Distortion Product Otoacoustic Emissions (DPOAEs), collected in the Department of Speech and Hearing Sciences, OSU. The logistic regression model and the noncentral F mixed effects regression models are explored to capture the important features of hearing data. In particular, noncentral-F mixed effects regression models capture within-subject-variability of the distortion products of healthy hearing subjects. This is key to understanding the underlying processes inherent in DPOAE-based hearing tests. The Penalized Quasi-Likelihood method is used to estimate the model parameters and we demonstrate how to do the model selection and diagnosing. |
