Error Variance Estimation in Ultrahigh-Dimensional Additive Models

Error variance estimation plays an important role in statistical inference for high-dimensional regression models. This article concerns with error variance estimation in high-dimensional sparse additive model. We study the asymptotic behavior of the traditional mean squared errors, the naive estimate of error variance, and show that it may significantly underestimate the error variance due to spurious correlations that are even higher in nonparametric models than linear models. We further propose an accurate estimate for error variance in ultrahigh-dimensional sparse additive model by effectively integrating sure independence screening and refitted cross-validation techniques. The root n consistency and the asymptotic normality of the resulting estimate are established. We conduct Monte Carlo simulation study to examine the finite sample performance of the newly proposed estimate. A real data example is used to illustrate the proposed methodology. Supplementary materials for this article are available online.