Smooth backfitting in generalized additive models

By Kyusang Yu, Byeong U. Park, Enno Mammen. 2008

95 citations recorded by the source corpus.

Generalized additive models have been popular among statisticians and data analysts in multivariate nonparametric regression with non-Gaussian responses including binary and count data. In this paper, a new likelihood approach for fitting generalized additive models is proposed. It aims to maximize a smoothed likelihood. The additive functions are estimated by solving a system of nonlinear integral equations. An iterative algorithm based on smooth backfitting is developed from the Newton--Kantorovich theorem. Asymptotic properties of the estimator and convergence of the algorithm are discussed. It is shown that our proposal based on local linear fit achieves the same bias and variance as the oracle estimator that uses knowledge of the other components. Numerical comparison with the recently proposed two-stage estimator [Ann. Statist. 32 (2004) 2412--2443] is also made.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/716deb19ac753a97c0db9c765c0766c9e0f9e20c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. H. Muller
2. Lijian Yang
3. Jane-ling Wang
4. Fang Yao
5. Yichao Wu
6. Byeong U. Park
7. Hua Liang
8. Li Wang

In Statistics and Probability

1. H. Muller
2. Lijian Yang
3. Jane-ling Wang
4. Byeong U. Park
5. Fang Yao
6. Yichao Wu
7. Hua Liang
8. Li Wang

Citations in the corpus

• Estimation and Inference for Generalized Geoadditive Models (2020, methodology): “…proposals for fitting the GAMs, for instance, the spline method of Stone (1986), Xue and Liang (2010), and Wang et al. (2011); the kernel method of Yu, Park, and Mammen (2008), Yang, Sperlich, and Härdle (2003), and Liang et al. (2008); and the two-stage methods of Horowitz and Mammen (2004),…”
• Varying Coefficient Regression Models: A Review and New Developments (2015, methodology): “If we take d D 1 with X1 1 and let the only index set I1 include all indices of other non-constant covariates, then model (3.3) reduces to the generalised additive model studied by Yu et al. (2008).”
• Time-Varying Additive Models for Longitudinal Data (2013, background): “…K denote a univariate kernel function, and for a bandwidth h define a boundary corrected kernel Kh(·; ·), as in Mammen, Linton, and Nielsen (1999) and Yu, Park, and Mammen (2008), by Kh(u; v) = K((u − v)/h)/h∫ I [K((x − v)/h)/h]dx · I (u, v ∈ I), where I is the support of the random variable to…”
• Continuously additive models for nonlinear functional regression (2013, methodology): “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).”
• Additive Functional Regression for Densities as Responses (2020, unknown): “A crucial element of the proof of P(‖T‖op ≤ γ ) → 1 as n → ∞ for some constant 0 < γ < 1 is the bound ∫ [0,1]2 ( p̂jk(xj, xk) p̂j(xj)pk(xk) − pjk(xj, xk) pj(xj)pk(xk) )2 pj(xj)pk(xk) dxj dxk = op(1); see Mammen, Linton, and Nielsen (1999), Yu, Park, and Mammen (2008), Lee, Mammen, and Park (2010), a”
• GENERALIZED ADDITIVE PARTIAL LINEAR MODELS WITH HIGH-DIMENSIONAL COVARIATES (2013, methodology): “In the early literature, kernel-based backfitting and local scoring procedures have been proposed by Buja, Hastie, and Tibshirani (1989) to iteratively estimate the linear coefficients and nonparametric components by solving a large system of equations (Yu, Park, and Mammen, 2008).”