Summary: A generalization of Sliced Inverse Regression to functional regressors was introduced by the authors [Statistics 37, No. 6, 475–488 (2003; Zbl 1032.62052
)]. Here we first address the issue of the identifiability of the Effective Dimension Reduction (EDR) space. Next, we estimate the covariance operator of the conditional expectation by means of kernel estimates. Consistency is proved and this extends the results of L.-X. Zhu
and K.-T. Fang
[Ann. Stat. 24, No. 3, 1053–1068 (1996; Zbl 0864.62027
)] in the multivariate context to the functional case. We also suggest a new way for estimating the EDR space for functional data which avoids inverting the covariance operator of the regressor. We apply our method to a prediction problem where the regressors are spectrometric curves.