## Adaptive estimation in time-series models.(English)Zbl 0941.62093

The authors focus their attention on adaptive estimation within a semi-parametric framework. The obtained results are well suited for time series models with i.i.d. errors. They consider the estimation of a Euclidean parameter $$\theta$$ in a semi-parametric model parametrized for this parameter and an infinite dimensional nuisance parameter $$g$$. In the time series models adaptive estimation of the Euclidean parameter is possible thanks to the independence of the present innovation and the past. The authors show that the necessary criteria for parameters to be adaptively estimable, given in a previous work and based on the specific structure of the score function in time series models, are also sufficient. Afterwards they built adaptive and hence efficient estimators and apply their results to ARMA and ARCH models.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F35 Robustness and adaptive procedures (parametric inference) 62G05 Nonparametric estimation 62F12 Asymptotic properties of parametric estimators

### Keywords:

LAN in time series; adaptive estimation; semiparametrics
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