Generalized functional linear models with semiparametric single-index interactions.

*(English)*Zbl 1392.62095Summary: We introduce a new class of functional generalized linear models, where the response is a scalar and some of the covariates are functional. We assume that the response depends on multiple covariates, a finite number of latent features in the functional predictor, and interaction between the two. To achieve parsimony, the interaction between the multiple covariates and the functional predictor is modeled semiparametrically with a single-index structure. We propose a two-step estimation procedure based on local estimating equations, and investigate two situations: (a) when the basis functions are predetermined, e.g., Fourier or wavelet basis functions and the functional features of interest are known; and (b) when the basis functions are data driven, such as with functional principal components. Asymptotic properties are developed. Notably, we show that when the functional features are data driven, the parameter estimates have an increased asymptotic variance due to the estimation error of the
basis functions. Our methods are illustrated with a simulation study and applied to an empirical dataset where a previously unknown interaction is detected.

##### MSC:

62G05 | Nonparametric estimation |

62G07 | Density estimation |

62H25 | Factor analysis and principal components; correspondence analysis |

62J12 | Generalized linear models (logistic models) |