Detecting Periodicity of a General Stationary Time Series via AR(2)-Model Fitting

TL;DR

This paper analyzes the effectiveness of fitting AR(2) models to stationary time series for detecting dominant spectral frequencies, showing that under certain spectral conditions, the method accurately identifies the true frequency with a convergence rate near n^{-2/3}.

Contribution

The paper provides theoretical insights into the consistency and convergence rate of AR(2)-based frequency estimators for stationary processes with spectral peaks.

Findings

AR(2) models can correctly identify dominant frequencies under sharp spectral peaks.

The estimator's convergence rate can approach n^{-2/3}, faster than the standard parametric rate.

Theoretical conditions for spectral peak sharpness ensure estimator consistency.

Abstract

Estimating the periodicity of a stationary time series via fitting a second order stationary autoregressive (AR(2)) model has been initiated by the seminal paper of Yule(1927).. We investigate properties of this procedure when applied to a general stationary processes possessing a spectral density with a dominant peak at some frequency $\lambda_0\in(0,\pi)$. We show that if the peak of the spectral density is sharp enough (in a way to be specified) then the AR(2) model, which best (in mean square sense) approximates the underlying process, correctly identifies the frequency $\lambda_0$. To investigate consistency properties of the AR(2) based estimator of $\lambda_0$, a near to pole framework is adopted. Triangular arrays of stationary stochastic processes are considered that possess a spectral density the peak of which at $\lambda_0$ becomes more pronounced as the sample size $n$ of the observed time series increases to infinity. It is shown in this set up, that the AR(2) based estimator achieves a rate of convergence which is larger than the parametric $n^{-1/2}$ rate and which can be arbitrarily close to $ n^{-2/3}$, the best rate that can be achieved by this estimator.