Prediction of weakly locally stationary processes by auto-regression
Résumé
In this contribution we introduce weakly locally stationary time series
through the local approximation of the non-stationary covariance structure by
a stationary one. This allows us to define autoregression coefficients in a
non-stationary context, which, in the particular case of a locally stationary
Time Varying Autoregressive (TVAR) process, coincide with the generating
coefficients. We provide and study an estimator of the time varying
autoregression coefficients in a general setting. The proposed estimator of
these coefficients enjoys an optimal minimax convergence rate under limited
smoothness conditions. In a second step, using a bias reduction technique, we
derive a minimax-rate estimator for arbitrarily smooth time-evolving
coefficients, which outperforms the previous one for large data sets. In
turn, for TVAR processes, the predictor derived from the estimator exhibits
an optimal minimax prediction rate.
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