A Note On The Adaptive Estimation Of A Quadratic Functional From Dependent Observations

Abstract

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.