Generalized partially functional linear model

In this paper, we propose a generalized functional linear regression model with scalar and functional multiple predictors. We develop maximum likelihood estimators for the regression coefficients. For the functional predictors, we adopt the method of functional principal component analysis to reduce their dimensions. We then propose the generalized auto-covariance operator, based on which an appropriate measure quantifies the difference between the estimators and their true values is established. The asymptotic joint distribution of estimated regression functions is proved. For the scalar predictors, we establish a distance between the estimated value and the true value, and prove the asymptotic property of the estimated regression coefficients. Extensive simulation experiment results are consistent with the theoretical result. Finally, two application examples of the model are given. One is sleep quality study where we studied the effects of heart rate, percentage of sleep time on total sleep in bed, wake after sleep onset and number of wakening during the night on sleep quality in 22 healthy people. The other one is mortality rate where we studied the effects of air quality index, temperature, relative humidity , GDP per capita and the number of beds per thousand people on the mortality rate across 80 major cities in China.