[Paper Review] Tests for comparing time-invariant and time-varying spectra based on the Anderson-Darling statistic

Shibin Zhang, Xin Tu|arXiv (Cornell University)|May 13, 2017

Spectroscopy and Chemometric Analyses17 references3 citations

TL;DR

This paper proposes two Anderson-Darling-based tests for comparing time-invariant and time-varying spectral densities of univariate time series using periodogram ratios. The tests are distribution-free, invariant to numerator-denominator exchange, and outperform Pearson-like alternatives by being more powerful, especially in detecting tail differences due to the heavy-tailed F(2,2) reference distribution.

ABSTRACT

Based on periodogram-ratios of two univariate time series at different frequency points, two tests are proposed for comparing their spectra. One is an Anderson-Darling-like statistic for testing the equality of two time-invariant spectra. The other is the maximum of Anderson-Darling-like statistics for testing the equality of two spectra no matter that they are time-invariant and time-varying. Both of two tests are applicable for independent or dependent time series. Several simulation examples show that the proposed statistics outperform those that are also based on periodogram-ratios but constructed by the Pearson-like statistics.

Motivation & Objective

• To address the limitation of Pearson-like goodness-of-fit tests for spectral comparison, which depend on arbitrary partitioning of the sample space.
• To develop a more powerful and invariant test for comparing spectral densities of time series.
• To extend the approach to both time-invariant and locally stationary time-varying spectra.
• To leverage the sensitivity of the Anderson-Darling statistic to tail differences in the F(2,2) distribution for improved detection of spectral discrepancies.
• To provide a computationally feasible and asymptotically valid test framework applicable to independent or dependent time series.

Proposed method

• The method uses periodogram ratios of two univariate time series at different frequencies as test statistics.
• For time-invariant spectra, an Anderson-Darling-like statistic is constructed to test whether periodogram ratios follow an F(2,2) distribution.
• For time-varying spectra, the maximum of Anderson-Darling-like statistics across frequency bands is used, leveraging blocking to handle local stationarity.
• The test is invariant under exchange of numerator and denominator in the periodogram ratio, ensuring robustness.
• Asymptotic distribution theory is derived under the null hypothesis of equal spectra, using empirical process and weak convergence arguments.
• The approach is validated through simulation studies comparing power against Pearson-like statistics.

Experimental results

Research questions

• RQ1 Can an Anderson-Darling-based test outperform Pearson-like statistics in detecting differences in time-invariant spectral densities?
• RQ2 How can the Anderson-Darling statistic be adapted to test for equality of time-varying spectra in locally stationary processes?
• RQ3 Does the invariance of the test under periodogram ratio exchange improve robustness and power?
• RQ4 To what extent does the heavy-tailed F(2,2) distribution enhance sensitivity to spectral differences in the tails?
• RQ5 How does the proposed test perform under dependence structures in time series data?

Key findings

• The proposed Anderson-Darling-based test for time-invariant spectra shows superior power compared to Pearson-like statistics, especially in detecting differences in the tails of the spectral distribution.
• The test is invariant to the order of the periodogram ratio, ensuring consistent results regardless of which series is in the numerator or denominator.
• The maximum of Anderson-Darling statistics across frequency bands effectively detects local spectral differences in time-varying processes.
• The asymptotic null distribution of the test statistics is derived and shown to be consistent under weak dependence and local stationarity.
• Simulation results confirm that the proposed test maintains correct size and exhibits higher power than competing methods, particularly when spectral differences are concentrated in the tails.
• The method remains valid for both independent and dependent time series, extending its applicability to real-world data.