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 pre-determined, 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 data set, where a previously unknown interaction is detected. Technical proofs of our theoretical results are provided in the online supplemental materials.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-CI-016-04
Acknowledgements: Yehua Li is Assistant Professor, Department of Statistics, University of Georgia, Athens, GA 30602 (E-mail: firstname.lastname@example.org). Naisyin Wang is Professor, Department of Statistics, University of Michigan, Ann Arbor, MI 48109-1107 (E-mail: email@example.com). Raymond J. Carroll is Distinguished Professor of Statistics, Nutrition and Toxicology, Department of Statistics, Texas A&M University, TAMU 3143, College Station, TX 77843-3143 (E-mail: firstname.lastname@example.org). Li's research was supported by the National Science Foundation (DMS-0806131). Wang's research was supported by a grant from the National Cancer Institute (CA74552). Carroll's research was supported by a grant from the National Cancer Institute (CA57030) and by award number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.