Consistency of trigonometric and polynomial regression estimators
Tom 25 / 1998
Streszczenie
The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials , k=0,1,..., for the observation model y_i = f(x_i) + η_i , i=1,...,n, where the η_i are independent random variables with zero mean value and finite variance, and the observation points x_i\in[a,b], i=1,...,n, form a random sample from a distribution with density ϱ\in L^1[a,b]. Sufficient and necessary conditions are obtained for consistency in the sense of the errors \Vert f-\widehat f_N\Vert, \vert f(x)-\widehatf_N(x)\vert, x\in[a,b], and E\Vert f-\widehatf_N\Vert^2 of the projection estimator \widehat f_N(x) = \sum_{k=0}^N\widehat{c}_ke_k(x) for \widehat{c}_0,\widehat{c}_1,\ldots,\widehat{c}_N determined by the least squares method and f\in L^2[a,b].