Fourier and Hermite series estimates of regression functions

Summary

In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type${{\hat m\left( x \right) = \sum\limits_{j = 1}^n {Y_{j\varphi N} } \left( {x,X_j } \right)} \mathord{\left/ {\vphantom {{\hat m\left( x \right) = \sum\limits_{j = 1}^n {Y_{j\varphi N} } \left( {x,X_j } \right)} {\sum\limits_{j = 1}^n {\varphi _N } \left( {x,X_j } \right)}}} \right. \kern-\nulldelimiterspace} {\sum\limits_{j = 1}^n {\varphi _N } \left( {x,X_j } \right)}}$, whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for$\hat m\left( x \right)$ to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then$\hat m\left( x \right)$ converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.