Some uniform consistency results in the partially linear additive model components estimation

In the present paper, we are mainly concerned with the partially linear additive model defined, for a measurable function psi(gamma(1)):= y(i) = Z(i)(T) beta + Sigma m(l) (X-li) + epsilon(i) <= i greater than or less than <= n. where Z(i) = (Z(i,1),..., Z(ip))(T) and X-i = (X-l,X-i,..., X-id)(T) are vectors of sigma(epsilon)and E(epsilon vertical bar X,Z) = 0 a.s. We establish exact rates of strong uniform consistency of the non linear additive components of the model estimated by the marginal integration device with the kernel method. Our proofs are based upon the modern empirical process theory in the spirit of the works of Einmahl and Mason (2000) and Deheuvels and Mason (2004) relative to uniform deviations of non parametric kernel-type estimators.