In this paper, we prove that functional sliced inverse regression (FSIR) achieves the optimal (minimax) rate for estimating the central space in functional sufficient dimension reduction problems. First, we provide a concentration inequality for the FSIR estimator of the covariance of the conditional mean, i.e., $\var(\E[\boldsymbol{X}\mid Y])$. Based on this inequality, we establish the root-$n$ consistency of the FSIR estimator of the image of $\var(\E[\boldsymbol{X}\mid Y])$. Second, we apply the most widely used truncated scheme to estimate the inverse of the covariance operator and identify the truncation parameter which ensures that FSIR can achieve the optimal minimax convergence rate for estimating the central space. Finally, we conduct simulations to demonstrate the optimal choice of truncation parameter and the estimation efficiency of FSIR. To the best of our knowledge, this is the first paper to rigorously prove the minimax optimality of FSIR in estimating the central space for multiple-index models and general $Y$ (not necessarily discrete).

翻译：在本文中，我们证明了函数型切片逆回归（FSIR）在函数型充分降维问题中能够达到估计中心空间的最优（极小极大）速率。首先，我们给出了条件均值协方差（即 $\var(\E[\boldsymbol{X}\mid Y])$）的FSIR估计量的浓度不等式。基于该不等式，我们建立了 $\var(\E[\boldsymbol{X}\mid Y])$ 像空间的FSIR估计量的根$n$相合性。其次，我们采用最广泛使用的截断方案来估计协方差算子的逆，并确定了使FSIR能够达到中心空间估计最优极小极大收敛速率的截断参数。最后，我们通过仿真实验展示了截断参数的最优选择以及FSIR的估计效率。据我们所知，这是首篇严格证明FSIR在多指标模型和一般$Y$（不限于离散型）情形下估计中心空间时具有极小极大最优性的论文。