Fourier and Hermite series estimates of regression functions

Abstract

In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X1,Y1),…, (Xn, Yn) 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 andϕN 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.