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 ψ:Rq→R$\psi : \mathbb {R}^{q} \rightarrow \mathbb {R}$, by ψ(Yi):=Yi=Zi⊤β+∑ℓ=1dmℓ(Xℓ,i)+ϵifor1≤i≤n, \begin{eqnarray*} \psi (\mathbf {Y}_{i}):=\mathcal {Y}_i=\mathbf {Z}^\top _i {\bm \beta } + \sum _{\ell =1}^{d}m_{\ell }(X_{\ell , i}) + \varepsilon _i\, \mbox{for}\,1\le i \le n, \end{eqnarray*} where Zi = (Zi, 1, …, Zip)⊤ and Xi = (X1, i, …, Xid)⊤ are vectors of explanatory variables, β=(β1,...,βp)⊤${\bm \beta } = (\beta _1,\ldots ,\beta _p)^\top$ is a vector of unknown parameters, m1, …, md are unknown univariate real functions, and ϵ1, …, ϵn are independent random errors with mean zero, finite variances σϵ, and E(ϵ|X,Z)=0$\mathbb {E}(\varepsilon |{\bf X}, {\bf Z} ) = 0$ 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.