Generalized Sliced Inverse Regression (GSIR) is one of the most important methods for nonlinear sufficient dimension reduction. As shown in Li and Song (2017), it enjoys a convergence rate that is independent of the dimension of the predictor, thus avoiding the curse of dimensionality. In this paper we establish an improved convergence rate of GSIR under additional mild eigenvalue decay rate and smoothness conditions. Our convergence rate can be made arbitrarily close to $n^{-1/3}$ under appropriate decay rate and smoothness parameters. As a comparison, the rate of Li and Song (2017) is $n^{-1/4}$ under the best conditions. This improvement is significant because, for example, in a semiparametric estimation problem involving an infinite-dimensional nuisance parameter, the convergence rate of the estimator of the nuisance parameter is often required to be faster than $n^{-1/4}$ to guarantee desired semiparametric properties such as asymptotic efficiency. This can be achieved by the improved convergence rate, but not by the original rate. The sharpened convergence rate can also be established for GSIR in more general settings, such as functional sufficient dimension reduction.

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