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

Christophe Chesneau (1) , Maher Kachour (2)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 LMJL - Laboratoire de Mathématiques Jean Leray

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 conditions.

Mots clés: Quadratic functional estimation; Plug-in approach; Wavelets; Rates of convergence; Mixing dependence

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Christophe Chesneau, Maher Kachour. A note on the adaptive estimation of a quadratic functional from dependent observations. 2012. ⟨hal-00707015v1⟩