Some uniform consistency results in the partially linear additive model components estimation
Abstract In the present paper, we are mainly concerned with the partially linear additive model defined, for a measurable function by where Zi (Zi, 1, Zip) and Xi (X1, i, Xid) are vectors of explanatory variables, is a vector of unknown parameters, m1, md are unknown univariate real functions, and 1, n are independent random errors with mean zero, finite variances s, and 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.