Variable Selection For Functional Linear Regression Models With Heredity Constraint

Functional data analysis has been a hot topic in the field of statistics in recent years.The basic idea behind functional data analysis is to express each individual in repeatedly measured data as a smooth function,which relaxes the structural constraints and distribution settings for data collection.Functional data analysis has received considerable attentions in various fields,including,for example,growth analysis,biomedicine,environmental science,economics and finance.In this paper,we mainly introduce variable selection for multiple functional linear regression model with strong heredity constraint.The strong heredity constraint here means that if there is an interaction term in the model,then its corresponding main effects must exist.We propose a new variable selection method to select the important functional predictors and interactions.Firstly,FPCA is used to estimate the functional principal component scores of the functional predictors,and the model is transformed into a high dimensional linear regression model with main terms and interaction terms of the functional principal component scores.Then,variable selection and parameter estimation of the linear regression model are carried out based on the group SCAD double penalty function.With appropriate selection of the tuning parameters,we establish the convergence rates of estimators for the coefficient functions and the consistency of the variable under some regularity conditions.Finally,simulation studies and Hong Kong environmental data analysis demonstrate the effectiveness of our proposed method on finite sample cases.