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

Christophe Chesneau (1) , Maher Kachour (2) , Fabien Navarro (3, 1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 Equipe Image - Laboratoire GREYC - UMR6072

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.

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

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

Christophe Chesneau, Maher Kachour, Fabien Navarro. A note on the adaptive estimation of a quadratic functional from dependent observations. Istatistik. Journal of the Turkish Statistical Association, 2013, 6 (1), pp.10--26. ⟨hal-00707015v2⟩