## 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

### Keywords:

exact asymptotics of minimax risk
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### References:

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