Convergence of non-linear functionals of smoothed empirical processes and kernel density estimates

José R. León (1) , Corinne Berzin (2) , José Rafael León (1) , Joaquín Ortega (3, 1)

1 UCV - Universidad Central de Venezuela
2 UPMF - Université Pierre Mendès France - Grenoble 2
3 IVIC - Instituto Venezolano de Investigaciones Cientificas = Venezuelan Institute for Scientific Research

We consider regularizations by convolution of the empirical process and study the asymptotic behaviour of nonlinear functionals of this process. Using a result for the same type of non-linear functionals of the Brownian bridge, shown in a previous paper [4], and a strong approximation theorem, we prove several results for the p-deviation in estimation of the derivatives of the density. We also study the asymptotic behaviour of the number of crossings of the smoothed empirical process defined by Yukich [17] and of a modified version of the Kullback deviation

Mots clés: Empirical process; Kernel density estimation; Kullback-Leibler deviation; Crossings; regularization by convolution; Non-linear functionals

Domaines: Probabilités [math.PR]

José R. León, Corinne Berzin, José Rafael León, Joaquín Ortega. Convergence of non-linear functionals of smoothed empirical processes and kernel density estimates. Statistics, 2003, 37 (3), pp.217-242. ⟨10.1080/02331880290015440⟩. ⟨hal-00319118⟩