In a multivariate nonparametric regression setting we construct explicit asymptotic uniform confidence bands for centered purely random forests. Since the most popular example in this class of random forests, namely the uniformly centered purely random forests, is well known to suffer from suboptimal rates, we propose a new type of purely random forests, called the Ehrenfest centered purely random forests, which achieve minimax optimal rates. Our main confidence band theorem applies to both random forests. The proof is based on an interpretation of random forests as generalized U-Statistics together with a Gaussian approximation of the supremum of empirical processes. Our theoretical findings are illustrated in simulation examples.

翻译：在多变量非参数回归框架下，我们为中心化纯随机森林构建了显式的渐近一致置信带。由于该类随机森林中最流行的示例——均匀中心化纯随机森林——已知存在次优收敛速率问题，我们提出一种新型纯随机森林，称为埃伦费斯特中心化纯随机森林，其能达到极小极大最优速率。我们的主要置信带定理同时适用于这两种随机森林。证明基于将随机森林解释为广义U统计量，并结合经验过程上确界的高斯逼近方法。理论结果通过仿真算例进行了验证。