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...le implementations of the functional additive model. While complex iterative procedures are required to fit a regular additive model (backfitting and variants, see, e.g., Hastie and Tibshirani, 1990; =-=Mammen and Park, 2005-=-), representations (10), (11) motivate a straightforward estimation scheme to recover the component functions fk, respectively fkm, by a series of one-dimensional smoothing steps. This not only leads ...

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... a modification of backfitting that works more reliably in the case of many components and irregular design and that allows a complete asymptotic theory. Nielsen and Sperlich [31] and Mammen and Park =-=[24, 25]-=- discuss practical implementation of smooth backfitting. Tjøstheim and Auestad [38], Linton and Nielsen [21] and Fan, Härdle and Mammen [9] discuss marginal integration estimators. See Christopeit and...

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... [34] and Opsomer [35] for the classical backfitting and in Mammen, Linton and Nielsen [28] for the smooth backfitting. Bandwidth choice and practical implementations are discussed in Mammen and Park =-=[29, 30]-=- and Nielsen and Sperlich [33]. The basic difference between smooth backfitting and backfitting lies in the fact that smooth backfitting is based on a smoothed least squares criterion whereas in the c...

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...ee [10] for a discussion on this. Mammen and Nielsen [17] considered a general class of nonlinear regression and discussed some estimation principles including the smooth backfitting. Mammen and Park =-=[18]-=- proposed several bandwidth selection methods for smooth backfitting, and Mammen and Park [19] provided a simplified version of the local linear smooth backfitting estimator in additive models.4 K. Y...

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...itting also achieves the oracle bounds. This has been shown for smooth backfitting estimates based on local linear fitting (see [9]). For practical implementations of smooth backfitting, see [12] and =-=[11]-=-. SomeA SIMPLE BACKFITTING METHOD 3 two-step procedures have been proposed for additive models. Christopeit and Hoderlein [2] use local quasi-differencing in the second step, an idea coming from effi...

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...9). Additive models (Friedman & Stuetzle 1981; Stone 1985) have been successfully used for many regression situations that involve continuous predictors and both continuous and generalized responses (=-=Mammen & Park 2005-=-; Yu et al. 2008; Carroll et al. 2008). The functional predictors we consider in the proposed time-additive approach to functional regression are assumed to be observed on their entire domain, usually...

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...lowing result holds for the asymptotic distribution of each component function fj(x j), j = 1, . . . , d: √ nhj ( fj(x j)− fj(xj)− βj(xj) ) d−→ N ( 0, ∫ K2(u) du σ2j (x j) pXj (xj) ) . (2.7) Here the asymptotic bias terms βj(x j) are defined as minimizers of (β1, . . . , βd) ∫ [β(x)− β1(x1)− · · · − βd(xd)]2pX(x) dx under the constraint that∫ βj(x j)pXj (x j)dxj = 1 2 h2j ∫ [2f ′j(x j)p′Xj (x j) + f ′′j (x j)pXj (x j)] dxj ∫ u2K(u) du, (2.8) where pX is the joint density of X = (X 1, . . . , Xd) and β(x) = 1 2 d∑ j=1 h2j [ 2f ′j(x j) ∂ log pX ∂xj (x) + f ′′j (x j) ] ∫ u2K(u) du. In [37] and [40] this asymptotic statement has been proved for the case that fj is estimated on a compact interval Ij . The conditions include a boundary modification of the kernel. Specifically, the convolution kernel h−1j K(h −1 j (X j i − xj)) is replaced by Khj (X j i , x j) = h−1j K(h −1 j (X j i − xj))/ ∫ Ij h−1j K(h −1 j (X j i − uj)) duj . Then it holds that ∫ Ij Khj (X j i , x j) dxj = 1. In particular, this implies ∫ Ij pXj ,Xk(x j , xk)dxj = pXk(x k) and ∫ Ij pXj (x j)dxj = 1 if one replaces h−1j K(h −1 j (X j i −xj)) by Khj (X j i , x j) in the definitions of the kernel density estimators. In f...