Robust Estimation Techniques for Stationary Processes via Nonlinear Periodogram Functionals

M. Felix1
  • 1

    Research Institute for Statistics and Information Science, University of Geneva, Switerland [manon.felix@unige.ch]

Keywords: Periodogram – Frequency domain estimation – M-estimation – Robust estimation

1 Abstract

In many scientific fields, the analysis of time series data often encounters the issue of additive contamination. This paper offers a comprehensive methodology to address this inference challenge. We propose a frequency-domain strategy based on weighted M-estimation, utilizing (potentially) nonlinear functionals of the periodogram ordinates, suitable for both linear and nonlinear stationary processes. The main advantages of this method are twofold. First, it bypasses the need for a parametric estimation of the contamination and subsequent filtering out of the process. Second, the estimation procedure employs directly the raw periodogram ordinates of the observed process, avoiding the need for a plug-in estimator. We prove the consistency of the resulting estimators (for both linear and nonlinear processes) and characterise their asymptotic distribution (asymptotic normality, for linear processes). The efficacy of this approach is demonstrated through Monte Carlo simulations and a real-world application involving a classification problem using EEG data from healthy and schizophrenic teenagers. This method combines time series analysis with machine learning techniques for potential schizophrenia diagnosis, outperforming traditional methods in terms of accuracy.

References

  • Beran [1994] Jan Beran. Statistics for Long-Memory Processes, volume 61. CRC Press, 1994.
  • Chiu [1990] Shean-Tsong Chiu. Peak-insensitive parametric spectrum estimation. Stochastic processes and their applications, 35(1):121–140, 1990.
  • Iacone [2010] Fabrizio Iacone. Local whittle estimation of the memory parameter in presence of deterministic components. Journal of Time Series Analysis, 31(1):37–49, 2010.
  • McCloskey and Hill [2017] Adam McCloskey and Jonathan B Hill. Parameter estimation robust to low-frequency contamination. Journal of Business & Economic Statistics, 35(4):598–610, 2017.
  • McCloskey and Perron [2013] Adam McCloskey and Pierre Perron. Memory parameter estimation in the presence of level shifts and deterministic trends. Econometric Theory, 29(6):1196–1237, 2013.
  • Shao and Wu [2007] Xiaofeng Shao and Wei Biao Wu. Asymptotic spectral theory for nonlinear time series. The Annals of Statistics, 35(4):1773–1801, 2007.
  • Taniguchi and Kakizawa [2012] Masanobu Taniguchi and Yoshihide Kakizawa. Asymptotic theory of statistical inference for time series. Springer Science & Business Media, 2012.
  • Von Sachs [1994] Rainer Von Sachs. Peak-insensitive non-parametric spectrum estimation. Journal of time series analysis, 15(4):429–452, 1994.
  • Wu [2011] Wei Biao Wu. Asymptotic theory for stationary processes. Statistics and its Interface, 4(2):207–226, 2011.
  • Wu and Min [2005] Wei Biao Wu and Wanli Min. On linear processes with dependent innovations. Stochastic Processes and their Applications, 115(6):939–958, 2005.