Date of Award

Fall 2012

Project Type


Program or Major

Mathematics: Statistics Concentration

Degree Name

Doctor of Philosophy

First Advisor

Linyuan Li


Test procedures for serial correlation of unknown form with wavelet methods are investigated in this dissertation. The new wavelet-based consistent test is motivated using Fan's (1996) canonical multivariate normal hypothesis testing model. In our framework, the test statistic relies on empirical wavelet coefficients of a wavelet-based spectral density estimator. We advocate the choice of the simple Haar wavelet function, since evidence demonstrates that the choice of the wavelet function is not critical. Under the null hypothesis of no serial correlation, the asymptotic distribution of a vector of empirical wavelet coefficients is derived, which is the multivariate normal distribution in the limit. It is also shown that the wavelet coefficients are asymptotically uncorrelated. The proposed test statistic presents the serious advantage to be completely data-driven or adaptive, which avoids the need to select any smoothing parameters. Furthermore, under a suitable class of local alternatives, the wavelet-based method is consistent against serial correlation of unknown form. The test statistic is expected to exhibit better power than the current test statistics when the true spectral density displays significant spatial inhomogeneity, such as seasonal or cycle periodicities. However, the convergence of the test statistic toward its respective asymptotic distribution is expected to be relatively slow. Thus, Monte Carlo methods are investigated to determine the corresponding critical value. In a small simulation study, the new method is compared with several current test statistics, with respect to their empirical levels and powers.