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A note on the adaptive estimation of a quadratic functional from dependent observations. (English) Zbl 1420.62171

Summary: We investigate the estimation of the integral of the square of a multidimensional unknown function \(f\) under mild assumptions on the model allowing dependence on the observations. We develop an adaptive estimator based on a plug-in approach and wavelet projections. Taking the mean absolute error and assuming that \(f\) has a certain degree of smoothness, we prove that our estimator attains a sharp rate of convergence. Applications are given for the biased density model, the nonparametric regression model and a GARCH-type model under some mixing dependence conditions \((\alpha\)-mixing or \(\beta\)-mixing). A simulation study considering nonparametric regression models with dependent observations illustrates the usefulness of the proposed estimator.

MSC:

62G08 Nonparametric regression and quantile regression
62G07 Density estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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