| The spectral analysis method is an important tool in time series analysis and the spectral density plays a crucial role on the spectral analysis. However, one of limitations of the spectral density is that the spectral density reflects only the covariance structure among several dependence measures in the time series data. To overcome this restriction, we define two spectral densities, the quantile spectral density and the association spectral density. The quantile spectral density can model the pairwise dependence structure and provide identification of nonlinear time series and the association spectral density allows detecting periodicities on different parts of the domain of the time series. We propose the estimators for the quantile spectral density and the association spectral density and derive their sampling properties including asymptotic normality. Furthermore, we use the quantile spectral density to develop a goodness-of-fit tests for time series and explain how this test can be used for comparing the sequential dependence structure of two time series. The asymptotic sampling properties of the test statistic is derived under the null and alternative hypothesis, and a bootstrap procedure is suggested to obtain finite sample approximation. The method is illustrated with simulations and some real data examples. Besides the exploration of the new spectral densities, we consider general quadratic forms of alpha-mixing time series and derive asymptotic normality of these forms under the relatively weak assumptions. |
