Efficient and fast spline-backfitted kernel smoothing of additive models

Abstract

A great deal of effort has been devoted to the inference of additive model in the last decade. Among existing procedures, the kernel type are too costly to implement for high dimensions or large sample sizes, while the spline type provide no asymptotic distribution or uniform convergence. We propose a one step backfitting estimator of the component function in an additive regression model, using spline estimators in the first stage followed by kernel/local linear estimators. Under weak conditions, the proposed estimator's pointwise distribution is asymptotically equivalent to an univariate kernel/local linear estimator, hence the dimension is effectively reduced to one at any point. This dimension reduction holds uniformly over an interval under assumptions of normal errors. Monte Carlo evidence supports the asymptotic results for dimensions ranging from low to very high, and sample sizes ranging from moderate to large. The proposed confidence band is applied to the Boston housing data for linearity diagnosis.

East African food security as influenced by future climate change and land use change at local to regional scales

“…RAMS was thoroughly tested and evaluated with recent observed data using NCEP forcings. Regionally specific fractional cover and leaf area index (LAI) estimates developed from MODIS imagery (Wang and Yang 2007) were incorporated to improve the regional atmospheric model's performance (Moore et al 2009). Since Indian Ocean temperatures may strongly influence coastal crop production (Funk et al 2008), we included monthly CCSM sea surface temperatures from the same scenario into RAMS and included a sizeable portion of the Indian Ocean in our domain.…”

Direct Simultaneous Inference in Additive Models and Its Application to Model Undernutrition

“…Härdle et al (2004) developed simultaneous confidence bands and specification tests for generalized additive models in the kernel regression context, and Wang and Yang (2009) for regression splines based on asymptotic considerations relying on a suboptimal choice of the number of knots. Härdle et al (2001) proposed locally adaptive (via wavelets) and bandwidth adaptive specification tests for additive models.…”

Oracally Efficient Two-Step Estimation of Generalized Additive Model

“…Our proof relies on "reducing bias by undersmoothing" and "averaging out the variance," accomplished with the joint asymptotics of kernel and spline functions for realizations of geometrically strongly mixing time series. These results are established under substantially greater technical difficulty than existing works on additive model such as Wang and Yang (2007), Wang and Yang (2009), Liu and Yang (2010), Ma and Yang (2011), and Song and Yang (2010). The additional complication is due to the lack of decomposition of spline estimation error into the sum of a bias and a noise term when the link function (b ) −1 is nonlinear.…”