Une mesure de la déviation quadratique d'estimateurs non paramétriques. (A measure of quadratic deviation of nonparametric estimators) (Q1080581)

Let \(\theta\) be either the density f of \(U_ 1\) or the regression function \(r=E(V_ 1| U_ 1)\) or the product rf and let \({\hat \theta}{}_ n\) be a kernel-type or an orthogonal series estimator of \(\theta\) based on n identically distributed \({\mathbb{R}}^ d\)-valued random variables \((U_ i,V_ i)_{1\leq i\leq n}\). Nonrandom sequences \((a_ n)_{n\in {\mathbb{N}}^*}\) and \((b_ n)_{n\in {\mathbb{N}}^*}\) are determined such that the statistics \[ (a_ n\int | {\hat \theta}_ n-\theta | d\mu -b_ n)_{n\in {\mathbb{N}}^*}, \] converge to a Gaussian distribution \(N(0,\sigma^ 2)\) (\(\mu\) is positive, \(\sigma\)- finite and absolutely continuous w.r.t. the Lebesgue measure). Both independent and mixing case are dealt with. It must be noticed that \(a_ n\), \(b_ n\) and \(\sigma\) do not depend on the mixing function. The basic tools of the paper are Gaussian approximations in Hilbert spaces and the Karhunen-Loeve expansion.