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A note on orthogonal series regression function estimators. (English) Zbl 0992.62039

Summary: The problem of non-parametric estimation of the regression function \(f(x)= E(Y |X=x)\) using the orthonormal system of trigonometric functions or Legendre polynomials \(e_k\), \(k=0,1,2,\ldots,\) is considered in the case where a sample of i.i.d. copies \((X_i,Y_i)\), \(i=1,\ldots,n,\) of the random variable \((X,Y)\) is available and the marginal distribution of \(X\) has density \(\varrho\in L^1[a,b]\).
The constructed estimators are of the form \(\widehat f_n(x) = \sum_{k=0}^{N(n)}\widehat c_ke_k(x)\), where the coefficients \(\widehat c_0,\widehat c_1,\ldots,\widehat c_N\) are determined by minimizing the empirical risk \[ n^{-1}\sum_{i=1}^n(Y_i -\sum_{k=0}^Nc_ke_k(X_i))^2. \] Sufficient conditions for consistency of the estimators in the sense of the errors \(E_X|f(X)-\widehat f_n(X)|^2\) and \(n^{-1}\sum_{i=1}^nE(f(X_i)-\widehat f_n(X_i))^2\) are obtained.

MSC:

62G08 Nonparametric regression and quantile regression
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