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Adaptive prediction and estimation in linear regression with infinitely many parameters. (English) Zbl 1043.62076
Summary: The problem of adaptive prediction and estimation in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that is sharp asymptotically minimax adaptive over ellipsoids in \(\ell_2\). The method consists in an application of blockwise Stein’s rule with “weakly” geometrically increasing blocks to the penalized least squares fits of the first \(N\) coefficients. To prove the results, we develop oracle inequalities for a sequence model with correlated data.

MSC:
62M20 Inference from stochastic processes and prediction
62J05 Linear regression; mixed models
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G08 Nonparametric regression and quantile regression
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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