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Nonparametric trigonometric orthogonal regression estimation

Research paper by Nora Saadi, Smail Adjabi

Indexed on: 08 Oct '16Published on: 17 May '16Published in: Comptes Rendus Mathematique



Abstract

Eubank, Hart, and Speckman (1990) [2] have investigated the nonparametric trigonometric regression estimator. They assumed that the observation xixi points satisfy <img height="22" border="0" style="vertical-align:bottom" width="111" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X16300565-si2.gif">∫axiψ(s)ds=(1+i)n, i=1,...,ni=1,...,n, where ψ∈L1[a,b]ψ∈L1[a,b] is a density satisfying certain smoothness conditions, and in a work by E. Rafajłowicz (1987) [3], the observation points coincide with knots of numerical quadratures. The aim of the present work is to introduce a new estimator of the regression function based on trigonometric series, for fixed point designs different from the ones considered so far, under milder restrictions on the observation points. This seems to be important since it may be numerically difficult to determine exactly the points xixi satisfying the recent condition or the knots of appropriate numerical quadratures, especially when their number is large.