Recursive estimation of a locally stationary process

We consider the problem of estimating the parameters of a locally stationary autoregressive process. This approach models the time evolution of the spectral content of a time series by a [0,1] → ℝ^d x ℝ₊ mapping of d linear prediction coefficients and the innovation variance. The identification problem for this model fits the classical non-parametric curve estimation theory. In this contribution we focus on recursive estimators and more particularly on the LMS (least mean square) algorithm. This estimator is based on a stochastic gradient approach. A precise study of its asymptotic behavior is proposed. It turns out that this estimator achieves the minimax rate only in a limited range of smoothness classes. We propose a bias reduction method which allows to achieve this rate in a wider range of smoothness classes.