  Large samples have been generated routinely from various sources. Classic
statistical models, such as smoothing spline ANOVA models, are not well
equipped to analyze such large samples due to expensive computational costs. In
particular, the daunting computational costs of selecting smoothing parameters
render smoothing spline ANOVA models impractical. In this article, we develop
an asympirical, i.e., asymptotic and empirical, smoothing parameters selection
approach for smoothing spline ANOVA models in large samples. The idea of this
approach is to use asymptotic analysis to show that the optimal smoothing
parameter is a polynomial function of the sample size and an unknown constant.
The unknown constant is then estimated through empirical subsample
extrapolation. The proposed method significantly reduces the computational
costs of selecting smoothing parameters in high-dimensional and large samples.
We show smoothing parameters chosen by the proposed method tend to the optimal
smoothing parameters that minimise a specific risk function. In addition, the
estimator based on the proposed smoothing parameters achieves the optimal
convergence rate. Extensive simulation studies demonstrate the numerical
advantage of the proposed method over competing methods in terms of relative
efficacies and running time. On an application to molecular dynamics data with
nearly one million observations, the proposed method has the best prediction
performance.
