Asymmetric Huber Periodogram

Abstract

This paper introduces a novel spectral M-estimator, called the asymmetric Huber periodogram (AHP), as a generalization of the ordinary periodogram (PG), the quantile periodogram (QP), and the Huber periodogram (HP). The AHP is constructed via trigonometric asymmetric Huber regression (AHR), in which a specially designed loss function replaces the squared ${\ell }_2$ loss used to define the PG. Relative to the QP, the AHP can be more computationally efficient, and relative to the HP and PG, it provides a more comprehensive characterization by examining the data across the range of the asymmetry parameter. We establish the theoretical properties of the AHP and investigate its relationship with the corresponding asymmetric Huber spectrum (AHS). Building on the asymptotic theory, we develop confidence intervals (CIs) for the AHS and propose a Fisher-type test. Simulation studies and two applications further demonstrate the AHP's effectiveness in detecting hidden periodicities, robustness to outliers, and utility for time series clustering.