Trigonometric regression estimation for observations with additive and multiplicative errors

Waldemar Popiński

Communications in Statistics - Theory and Methods, 2016, vol. 45, issue 3, 804-812

Abstract: The problem of non parametric function fitting using the complete orthogonal system of trigonometric functions em, m = 0, ±1, ±2, …, for the observation model yj = djf(xjn) + ηj, j = 0, 1, …, n − 1, is considered, where f:[0,2π]→C$f:[0,2\pi ]\rightarrow \mathbb {C}$, ηj are uncorrelated random variables with zero mean value and finite variance, dj are uncorrelated random variables with mean value d ≠ 0 and finite variance, independent of ηj, and the observation points xjn ∈ [0, 2π] are equidistant. Conditions for convergence of the integrated mean-square error E∥f-f^N(n)∥22$E\Vert f-\hat{f}_{N(n)}\Vert ^2_2$ and the pointwise mean-square error E(f(x)-f^N(n)(x))2$E(f(x)-\hat{f}_{N(n)}(x))^2$ of the estimator f^N(n)(x)=∑m=-N(n)N(n)c˜mem(x)/d$\hat{f}_{N(n)}(x) = \sum _{m=-N(n)}^{N(n)}\tilde{c}_me_m(x)/d$ for f ∈ BV[0, 2π] and coefficients c˜-N(n),...,c˜N(n)$\tilde{c}_{-N(n)},\ldots,\tilde{c}_{N(n)}$ obtained by the least squares method are studied.

Date: 2016
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DOI: 10.1080/03610926.2013.851236

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