Quantile Fourier transform, quantile series, and nonparametric estimation of quantile spectra

Quantile spectra were recently introduced through trigonometric quantile regression as an alternative to the conventional power spectra for quantile-frequency analysis (QFA) of time series. As bivariate functions of frequency and quantile level, quantile spectra have found useful application in a number of scientific and engineering disciplines including financial time series analysis and signal processing. In this paper, a nonparametric method is proposed for estimating quantile spectra as bivariate functions of frequency and quantile level. This method is based on the quantile discrete Fourier transform (QDFT), derived from trigonometric quantile regression, and the resulting quantile series (QSER) as the inverse Fourier transform of the QDFT. The construction of the QSER enables an application of the lag-window (LW) approach to quantile spectrum estimation in a way similar to conventional spectrum estimation. In addition, a post-processing step of smoothing across quantiles is employed to further reduce the statistical variability of the LW estimator when the underlying spectrum varies smoothly with respect to the quantile level. This paper provides the results of a simulation study to evaluate the proposed method.