A Nonlinear Method for Robust Spectral Analysis

A nonlinear spectral analyzer, called the
L

_p
-norm periodogram, is obtained by replacing the least-squares criterion with an
L

_p
-norm criterion in the regression formulation of the ordinary periodogram. In this paper, we study the statistical properties of the
L

_p
-norm periodogram for time series with continuous and mixed spectra. We derive the asymptotic distribution of the
L

_p
-norm periodogram and discover an important relationship with the so-called fractional autocorrelation spectrum that can be viewed as an alternative to the power spectrum in representing the serial dependence of a random process in the frequency domain. In comparison with the ordinary periodogram
(p
= 2), we show that by varying the value of
p
in the interval (1,2) the
L

_p
-norm periodogram can strike a balance between robustness against heavy-tailed noise, efficiency under regular conditions, and spectral leakage for time series with mixed spectra. We also show that the
L

_p
-norm periodogram can detect serial dependence of uncorrelated non-Gaussian time series that cannot be detected by the ordinary periodogram.