On Recursive Estimation for Time Varying Autoregressive Processes

• Published in: The Annals of statistics Vol. 33; no. 6; pp. 2610 - 2654
• Main Authors: Moulines, Eric, Priouret, Pierre, Roueff, François
• Format: Journal Article
• Language: English
• Published: Hayward, CA Institute of Mathematical Statistics 01-12-2005; The Institute of Mathematical Statistics
• Subjects: 60J27; 62G08; 62G20; 62M10; Approximation; Autoregressive models; Covariance matrices; Difference equations; Estimating techniques; Estimators; Exact sciences and technology; Inference from stochastic processes; time series analysis; Integers; Locally stationary processes; Markov processes; Mathematical models; Mathematical vectors; Mathematics; Minimax; nonparametric estimation; Nonparametric inference; Polynomials; Probability and statistics; Probability theory and stochastic processes; Random variables; recursive estimation; Sciences and techniques of general use; Statistical analysis; Statistics; Studies; Testing and Estimation with Stochastic Process Data; time-varying autoregressive model; Statistical method; Estimation error; Reduction method; Stability; Asymptotic expansion; Convergence rate; Non parametric estimation; Statistical estimation; Time variation; Biased estimation; Recursive estimation; Autoregressive processes
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/009053605000000624

This paper focuses on recursive estimation of time varying autoregressive processes in a nonparametric setting. The stability of the model is revisited and uniform results are provided when the time-varying autoregressive parameters belong to appropriate smoothness classes. An adequate normalization for the correction term used in the recursive estimation procedure allows for very mild assumptions on the innovations distributions. The rate of convergence of the pointwise estimates is shown to be minimax in β-Lipschitz classes for 0 < β ≤ 1. For 1 < β ≤ 2, this property no longer holds. This can be seen by using an asymptotic expansion of the estimation error. A bias reduction method is then proposed for recovering the minimax rate.