Asymptotically optimal selection of a piecewise polynomial estimator of a regression function
Consider a random vector (X,Y) with values in \([0,1]^ d\times {\mathbb{R}}\) and assume that \(g(x)=E(Y| X=x)\) exists. Given an i.i.d sample \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) the regression function g has to be estimated. The approach discussed in this paper is the method of fitting piecewise polynomials of degree \(k_ n\) on small subcubes of length \(m_ n^{-1}\). The amount of smoothing is controlled by the parameters \((k_ n,m_ n)\) and these can be chosen by minimizing the final prediction error (FPE-criterion) of \textit{H. Akaike} [Ann. Inst. Stat. Math. 22, 203-217 (1970; Zbl 0259.62076)]. The author proves that this procedure is asymptotically efficient with respect to the squared error provided some moment conditions hold, the density of X is bounded from below and g is not a piecewise polynomial itself.
scientific article; zbMATH DE number 4030721
scientific article