We study linear smoothers and their use in building nonparametric regression models. In the first part of this paper we examine certain aspects of linear smoothers for scatterplots; examples of these are the running-mean and running-line, kernel and cubic spline smoothers. The eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother, and several other topics such as the number of degrees of freedom of a smoother are discussed. In the second part of the paper we describe how linear smoothers can be used to estimate the additive model, a powerful nonparametric regression model, using the "back-fitting algorithm." We show that backfitting is the Gauss-Seidel iterative method for solving a set of normal equations associated with the additive model. We provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers.
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A new method is presented for flexible regression modeling of high dimensional data. The model takes the form of an expansion in product spline basis functions, where the number...
Given data $z_i = g(t_i ) + \varepsilon _i , 1 \leqq i \leqq n$, where g is the unknown function, the $t_i $ are known d-dimensional variables in a domain $\Omega $, and the $\v...
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The problem of regression under Gaussian assumptions is treated generally. The relationship between Bayesian prediction, regularization and smoothing is elucidated. The ideal re...
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