Non-Asymptotic Analysis of Classical Spectrum Estimators for $L$-mixing Time-series Data with Estimated Means
Provides rigorous finite-sample guarantees for classical spectral estimators, benefiting time-series analysis in fields like economics and climatology.
The authors derive non-asymptotic error bounds of O(1/√k) for Bartlett and Welch spectral estimators for L-mixing time-series with unknown means, achieving the tightest bounds among existing work.
Spectral estimation is an important tool in time series analysis, with applications including economics, astronomy, and climatology. The asymptotic theory for non-parametric estimation is well-known but the development of non-asymptotic theory is still ongoing. Our recent work obtained the first non-asymptotic error bounds on the Bartlett and Welch methods with restrictive assumptions. In this work, we derive non-asymptotic error bounds for both Bartlett and Welch estimators for $L$-mixing time-series data with unknown means, and the results cover the special case with known zero means. The class of $L$-mixing processes contains common models in time series analysis, including autoregressive processes and measurements of geometrically ergodic Markov chains. Our new error bounds are of $O(\frac{1}{\sqrt{k}})$, where $k$ is the number of data segments used in the algorithm. Such bounds are the tightest among the existing work on non-asymptotic analysis of classical spectrum estimators with or without zero-mean assumptions.