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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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...ear 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. YU, B. U. PARK AND E. MAMMEN The rest of the paper is structured as follows. In Section 2 we in...

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...me asymptotic distribution for both estimators (up to an additive norming term). Asymptotically one does not lose any efficiency by not knowing the additive components fk : k 6= j compared to the oracle model where these components are known. This is an asymptotic optimality result for the local linear smooth backfitting. It achieves the same asymptotic bias and variance as in the oracle model. As discussed above, the Nadaraya-Watson smooth backfitting estimator achieves only the asymptotic variance of the oracle model. For an alternative implementation of local linear smooth backfitting, see [41]. 2.4 Smooth backfitting as solution of a noisy integral equation We write the smooth backfitting estimators as solutions of an integral equation. We discuss this briefly for Nadaraya-Watson smoothing. Put f(x1, . . . , xd) = (f1(x1), . . . , fd(xd)) > and f∗(x1, . . . , xd) = (f∗1 (x1), . . . , f ∗ d (xd)) >. With this notation we can rewrite (2.3) as f(x) = f∗(x)− ∫ H(x, z)f(z) dz, (2.15) where for each value of x, z ∈ R the integral kernel H(x, z) is a matrix with element (j, k) equal to pXj ,Xk(x j , xk)/pXj (x j). This representation motivates an alternative algorithm. One ca...

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...on of smoothing backfitting algorithms, including the one based on more efficient local linear estimation, we refer to Mammen et al. [15] and Nielsen and Sperlich [21]. More recently, Mammen and Park =-=[17]-=- showed that if in (2.12) ̂mj is replaced by a marginal local linear estimator, and ̂ fjℓ/ ̂ fj is replaced by a more sophisticated functional constructed using a convolution kernel, the resulting bac...

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...n the uncensored case, several methods have been proposed to estimate the additive regression function. We shall evoke, among others, the methods based on B-splines [36], on the backfitting algorithm =-=[21, 23, 32]-=- and on marginal integration [34, 40, 31]. In [17], Fan and Gijbels established the asymptotic normality for estimators obtained via the backfitting algorithm combined with various synthetic data esti...

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...man and Stuetzle 1981; Buja, Hastie and Tibshirani 1989; Opsomer and Ruppert 1998), the smooth backfitting algorithm (Mammen, Linton and Nielsen 1999, Mammen and Park 2005, Nielsen and Sperlich 2005; =-=Mammen and Park 2006-=-; Yu, Park and Mammen 2008), marginal integration estimation methods (Tjøstheim and Auestad 1994; Linton and Nielsen 1995; Fan et al. 1998), the Fourier series or wavelets approximation approach (Amat...