Nonparametric two sample test of spectral densities

A novel nonparametric test for the equality of the covariance matrices of two Gaussian stationary processes, possibly of different lengths, is proposed. The test translates to testing the equality of two spectral densities and is shown to be minimax rate-optimal. Test performance is validated in a simulation study, and the practical utility is demonstrated in the analysis of real electroencephalography data. The test is implemented in the R-package sdf.test.

💡 Research Summary

This paper introduces a fully non‑parametric two‑sample test for assessing whether two Gaussian stationary time series—potentially of different lengths—share the same covariance structure, equivalently whether their spectral density functions are identical. The authors build on the recent non‑parametric log‑spectral density estimator of Klockmann and Krivobokova (2024), which relies on a discrete cosine transform (DCT) and a binning procedure that equalises the effective sample size of the two series.

The methodology proceeds in four steps. First, each series is transformed by the DCT‑I matrix, producing approximately independent gamma‑distributed coefficients whose means are proportional to the underlying spectral density. Second, the coefficients are aggregated into a common number of bins (T) (chosen via exponents (\nu_1,\nu_2) that depend on the Hölder smoothness parameters), yielding binned observations (Q_{k,t}) for both series. Third, a variance‑stabilising transformation (logarithm followed by a square‑root) converts the binned data into approximately Gaussian observations (Y_{k,t}) with known variance (1/m_k) where (m_k=n_k/T). A deterministic centring step produces (Y^*_{k,t}), which Lemma 1 shows can be decomposed into a deterministic bias term, an i.i.d. Gaussian noise component, and a higher‑order remainder that is asymptotically negligible.

Fourth, the transformed data are used to estimate the functions (g_k(x)=2^{-1/2}\log f_k(x)) via periodic smoothing splines of order (2q-1). The spline estimator solves a penalised least‑squares problem with smoothing parameter (h_k). The final test statistic is the empirical (L_2) distance between the two spline estimates, \