Fourier and Hermite series estimates of regression functions (Q1091695)

Let (X,Y) be a pair of random variables; X has a density f. A regression function \(m(x)=E\{Y| X=x\}\) is first estimated by \[ \hat m(x)=\sum^{n}_{j=1}Y_ jD_ N(x-X_ j)/\sum^{n}_{j=1}D_ N(x- X_ j), \] where \(\pi D_ N\) is the Dirichlet kernel of N-th oder, \((X_ j,Y_ j)\) are observations of (X,Y), N depends on n (a number of observations). The authors examine two cases of convergence \(\hat m\to m:\) ''in probability'' and ''almost completely''. There are given general conditions for these types of convergence. Formulas for the convergence rates are also given. The second type of estimation is analogous. \(D_ N\) is replaced here by \(d_ N(x,X_ j)\), where \(d_ N(x,y)\) is defined using Hermite polynomials. The modified convergence conditions give analogous convergence results as above.