On the Optimality of Functional Sliced Inverse Regression

TL;DR

This paper proves that functional sliced inverse regression (FSIR) achieves the optimal minimax rate for estimating the central space in functional sufficient dimension reduction, providing theoretical guarantees and practical insights.

Contribution

It is the first to rigorously establish the minimax optimality of FSIR for estimating the central space in functional models, including non-discrete responses.

Findings

FSIR achieves root-n consistency in estimating the covariance of the conditional mean.

The paper identifies the optimal truncation parameter for the covariance operator.

Simulations confirm the theoretical optimality and efficiency of FSIR.

Abstract

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. Based on this inequality, we establish the root-$n$ consistency of the FSIR estimator of the image of covariance of the conditional mean. Second, we apply the most widely used truncated scheme to estimate the inverse of the covariance operator and identify the truncation parameter that 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).