Core Concepts
This paper presents a finite-time convergence analysis of the Bartlett and Welch spectral estimators for L-mixing time series data, demonstrating that the error bounds are determined by the data's L-mixing properties and match existing bounds in more restrictive settings up to logarithmic factors.
Abstract
Bibliographic Information:
Zheng, Y., & Lamperski, A. (2024). Nonasymptotic Analysis of Classical Spectrum Estimators with L-mixing Time-series Data. arXiv preprint arXiv:2410.02951.
Research Objective:
This paper aims to establish finite-time error bounds for the widely used Bartlett and Welch spectral estimators when applied to L-mixing time series data, a class of processes encompassing various models in time series analysis.
Methodology:
The authors leverage the theory of L-mixing processes to derive non-asymptotic error bounds for both the variance and bias of the Bartlett and Welch estimators. They extend classical L-mixing results to vector-valued processes and relate the L-mixing properties of the data sequences to the matrices used in spectral estimation.
Key Findings:
- The error bounds for both estimators are shown to be determined by the L-mixing properties of the data, specifically the mixing constant and moment bounds.
- The concentration bound for the Bartlett estimator is independent of the window length, while for the Welch estimator, it depends on the ratio between the window length and the data chunk size.
- The convergence rate of the algorithm is of order O(1/√k log2(log2 k)), where k represents the number of data chunks used.
- The derived error bounds match existing bounds derived under more restrictive assumptions (Gaussian or linear filter-based data) up to logarithmic factors.
Main Conclusions:
The study provides a rigorous finite-time convergence analysis for the Bartlett and Welch spectral estimators under the L-mixing assumption, demonstrating that these estimators achieve favorable error rates for a broad class of time series data.
Significance:
This work contributes significantly to the non-asymptotic theory of non-parametric spectral estimation, which has been less developed compared to its asymptotic counterpart or the non-asymptotic theory of parametric methods. The results are relevant for practical applications where only finite data records are available.
Limitations and Future Research:
- The current analysis assumes a zero-mean time series, which might not always hold in practice. Future work could extend the analysis to non-zero-mean time series.
- The derived error bounds, while theoretically sound, are acknowledged to be conservative. Further research could explore tighter bounds by refining the bounding techniques.
Stats
The Doeblin coefficient in the Markov chain example is δ = 0.72.
The upper bound of the L-mixing statistics is Γd,4q(y) ≤ 4Gmax/(δ^(4q)), where Gmax = maxk ∥y[k]∥.
The upper bound of the moment is M4q(y) ≤ Gmax.
The probability parameter in Theorem 2 is set to ν = 0.1, indicating a 90% probability for the theoretical bound to hold.
For the Bartlett estimator simulation, M = 5 and L = 10^7.
For the Welch estimator simulation, a Hann window is used with M = 16, K = 8, and L = 10^7.