A functional sliced inverse regression (FSIR) technique is considered for fitting the model $$ Y=f(\langle X,a_1\rangle,\dots,\langle X,a_p\rangle,\varepsilon), $$ where $Y$ is the response, $X$ is a functional regressor (a random function in $L_2[a,b]$), $f$ is an unknown regression function, $a_1$ are unknown functions, and $\varepsilon$ is an error term. Consistency of regularized FSIR estimates for $a_i$ is demonstrated. It is proposed to estimate the function $f$ by a multilayer perceptron technique. Consistency of the proposed training algorithm for such perceptrons is shown. Applications to real data are considered.
Reviewer: R. E. Maiboroda (Kyïv)