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

TL;DR

This paper introduces a nonparametric approach for estimating quantile spectra and cross-spectra using Fourier transforms and quantile regression, with smoothing techniques to improve estimation accuracy across quantiles.

Contribution

It develops a novel nonparametric spectral estimation method based on the quantile discrete Fourier transform and quantile series, enhancing spectral analysis at different quantile levels.

Findings

Effective in reducing variability with smoothing techniques

Performs well in simulation studies

Captures spectral features across quantiles

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.