We develop a novel method to construct uniformly valid confidence bands for a nonparametric component $f_1$ in the sparse additive model $Y=f_1(X_1)+\ldots + f_p(X_p) + \varepsilon$ in a high-dimensional setting. Our method integrates sieve estimation into a high-dimensional Z-estimation framework, facilitating the construction of uniformly valid confidence bands for the target component $f_1$. To form these confidence bands, we employ a multiplier bootstrap procedure. Additionally, we provide rates for the uniform lasso estimation in high dimensions, which may be of independent interest. Through simulation studies, we demonstrate that our proposed method delivers reliable results in terms of estimation and coverage, even in small samples.

翻译：我们提出了一种新颖的方法，用于在高维稀疏可加模型 $Y=f_1(X_1)+\ldots + f_p(X_p) + \varepsilon$ 中为非参数分量 $f_1$ 构建一致有效的置信带。该方法将筛估计融入高维Z估计框架，从而便于为目标分量 $f_1$ 构建一致有效的置信带。为形成这些置信带，我们采用了乘子自助法。此外，我们提供了高维均匀Lasso估计的收敛速率，这一结果可能具有独立的研究价值。通过模拟研究，我们证明了所提方法即使在样本量较小的情况下，也能在估计和覆盖概率方面提供可靠的结果。