  In this paper, a new estimation method is introduced for the quantile
spectrum, which uses a parametric form of the autoregressive (AR) spectrum
coupled with nonparametric smoothing. The method begins with quantile
periodograms which are constructed by trigonometric quantile regression at
different quantile levels, to represent the serial dependence of time series at
various quantiles. At each quantile level, we approximate the quantile spectrum
by a function in the form of an ordinary AR spectrum. In this model, we first
compute what we call the quantile autocovariance function (QACF) by the inverse
Fourier transformation of the quantile periodogram at each quantile level.
Then, we solve the Yule-Walker equations formed by the QACF to obtain the
quantile partial autocorrelation function (QPACF) and the scale parameter.
Finally, we smooth QPACF and the scale parameter across the quantile levels
using a nonparametric smoother, convert the smoothed QPACF to AR coefficients,
and obtain the AR spectral density function. Numerical results show that the
proposed method outperforms other conventional smoothing techniques. We take
advantage of the two-dimensional property of the estimators and train a
convolutional neural network (CNN) to classify smoothed quantile periodogram of
earthquake data and achieve a higher accuracy than a similar classifier using
ordinary periodograms.
