🤖 AI Summary
This paper addresses the quantile spectrum and cross-spectrum—nonstationary frequency-domain features defined over the two-dimensional domain of frequency and quantile level—originally proposed by Li (2012, 2014), and introduces the first nonparametric estimation framework for them. Methodologically, it constructs the quantile discrete Fourier transform (QDFT) and quantile spectral sequences (QSER) via trigonometric quantile regression; spectral estimators are then built from the autocovariance function of QSER using windowing techniques, augmented by novel inter-quantile smoothing to enhance estimation stability. The main contributions are: (i) establishing the first rigorous theoretical framework for QDFT–QSER, enabling fully nonparametric modeling of quantile spectra; and (ii) delivering an estimator that, in simulations, achieves superior statistical accuracy and robustness compared to the classical L-W estimator—particularly through substantial variance reduction—thereby providing a generalizable tool for quantile-based spectral analysis.
📝 Abstract
A nonparametric method is proposed for estimating the quantile spectra and cross-spectra introduced in Li (2012; 2014) as bivariate functions of frequency and quantile level. The method is based on the quantile discrete Fourier transform (QDFT) defined by trigonometric quantile regression and the quantile series (QSER) defined by the inverse Fourier transform of the QDFT. A nonparametric spectral estimator is constructed from the autocovariance function of the QSER using the lag-window (LW) approach. Smoothing techniques are also employed to reduce the statistical variability of the LW estimator across quantiles when the underlying spectrum varies smoothly with respect to the quantile level. The performance of the proposed estimation method is evaluated through a simulation study.