A note on orthogonal series regression function estimators

The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials ${\displaystyle e_k}$, k=0,1,2,..., is considered in the case where a sample of i.i.d. copies ${\displaystyle (X_i,Y_i)}$, i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ ${\displaystyle L^1}$[a,b]. The constructed estimators are of the form ${\displaystyle w{\hat{f}}_n\left(x\right)=\sum _{k=0}^{N\left(n\right)}w{\hat{c}}_k{e}_k\left(x\right)}$, where the coefficients ${\displaystyle w{\hat{c}}_0,w{\hat{c}}_1,...,w{\hat{c}}_N}$ are determined by minimizing the empirical risk ${\displaystyle n^{-1}\sum _{i=1}^n{(Y_i-\sum _{k=0}^N{c}_k{e}_k\left(X_i\right))}^2}$. Sufficient conditions for consistency of the estimators in the sense of the errors ${\displaystyle E_{X}vertf\left(X\right)-w{\hat{f}}_n\left(X\right)ver{t}^2}$ and ${\displaystyle n^{-1}\sum _{i=1}^{n}E{(f\left(X_i\right)-w{\hat{f}}_n\left(X_i\right))}^2}$ are obtained.