Quantile Method for Time Series
In this chapter, we introduce a nonparametric method to statistically investigate stationary time series. We have seen that there exists a spectral distribution function for any second-order stationary process. We define quantiles of the spectral distribution function in the frequency domain and consider the quantile method for parameter estimation of stationary time series. The estimation method for quantiles is generally formulated by minimizing a check function. The quantile estimator is shown to be asymptotically normal. We also consider the hypothesis testing problem for quantiles in the frequency domain and propose a test statistic associated with our quantile estimator, which asymptotically converges to the standard normal under the null hypothesis. The finite sample performance of the quantile estimator is shown in our numerical studies.