A note on the adaptive estimation of a quadratic functional from dependent observations.

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.