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A nonparametric method is proposed for estimating the quantile spectra andcross-spectra introduced in Li (2012; 2014) as bivariate functions of frequencyand quantile level. The method is based on the quantile discrete Fouriertransform (QDFT) defined by trigonometric quantile regression and the quantileseries (QSER) defined by the inverse Fourier transform of the QDFT. Anonparametric spectral estimator is constructed from the autocovariance andcross-covariance functions of the QSER using the lag-window (LW) approach.Various quantile smoothing techniques are employed further to reduce thestatistical variability of the estimator across quantiles, among which is a newtechnique called spline quantile regression (SQR). The performance of theproposed estimation method is evaluated through a simulation study.

Quantile Fourier Transform, Quantile Series, and Nonparametric Estimation of Quantile Spectra