Least-squares trigonometric regression estimation

Volume 26 / 1999

Waldemar Popiński Applicationes Mathematicae 26 (1999), 121-131 DOI: 10.4064/am-26-2-121-131


The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions $e_k$, k=0,1,2,..., for the observation model $y_i = f(x_{in}) + η_i$, i=1,...,n, is considered, where $η_i$ are uncorrelated random variables with zero mean value and finite variance, and the observation points $x_{in} ∈ [0,2π]$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1/n)\sum_{i=1}^n E(f(x_{in})-\widehat f_{N(n)}(x_{in}))^2$, the integrated mean-square error $E ‖f-\widehat f_{N(n)}‖^2$ and the pointwise mean-square error $E(f(x)-\widehatf_{N(n)}(x))^2$ of the estimator $\widehat f_{N(n)}(x) = \sum_{k=0}^{N(n)} \widehat c_k e_k(x)$ for f ∈ C[0,2π] and $\widehat c_0,\widehat c_1,...,\widehat c_{N(n)}$ obtained by the least squares method are studied.


  • Waldemar Popiński

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