A note on orthogonal series regression function estimators

Volume 26 / 1999

Waldemar Popiński Applicationes Mathematicae 26 (1999), 281-291 DOI: 10.4064/am-26-3-281-291

Abstract

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 $\widehat f_n(x) = \sum_{k=0}^{N(n)}\widehat c_ke_k(x)$, where the coefficients $\widehat c_0,\widehat c_1,...,\widehat c_N$ 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 $E_X\vert f(X)-\widehat f_n(X)\vert^2$ and $n^{-1}\sum_{i=1}^nE(f(X_i)-\widehat f_n(X_i))^2$ are obtained.

Authors

  • Waldemar Popiński

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