On least squares estimation of Fourier coefficients and of the regression function

Waldemar Popiński

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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 $e_{k}$ , k=0,1,2,..., is considered in the case where a sample of i.i.d. copies $(X_{i}, Y_{i})$ , i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ $L^{1}$ [a,b]. The constructed estimators are of the form ${\hat{f}}_{n} (x) = \sum_{k = 0}^{N (n)} {\hat{c}}_{k} e_{k} (x)$ , where the coefficients ${\hat{c}}_{0}, {\hat{c}}_{1}, . . ., {\hat{c}}_{N}$ are determined by minimizing the empirical risk $n^{- 1} \sum_{i = 1}^{n} {(Y_{i} - \sum_{k = 0}^{N} c_{k} e_{k} (X_{i}))}^{2}$ . Sufficient conditions...