On the estimation of density-weighted average derivative by wavelet methods under various dependence structures

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

The problem of estimating the density-weighted average derivative of a regression function is considered. We present a new consistent estimator based on a plug-in approach and wavelet projections. Its performances are explored under various dependence structures on the observations: the independent case, the $\rho$-mixing case and the $\alpha$-mixing case. More precisely, denoting $n$ the number of observations, in the independent case, we prove that it attains $1/n$ under the mean squared error, in the $\rho$-mixing case, $1/\sqrt{n}$ under the mean absolute error, and, in the $\alpha$-mixing case, $\sqrt{\ln n /n}$ under the mean absolute error. A short simulation study illustrates the theory.

Mots clés: Wavelets; Plug-in' approach; $\alpha$-mixing; Nonparametric estimation of density-weighted average derivative; $\alpha$-mixing.; $\rho$-mixing; Consistency; 'Plug-in' approach

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

Christophe Chesneau, Maher Kachour, Fabien Navarro. On the estimation of density-weighted average derivative by wavelet methods under various dependence structures. Sankhya A, 2014, ⟨10.1007/s13171-013-0032-1⟩. ⟨hal-00668544v3⟩