Statistical Inference For Several Functional Mixed Effects Models

With the development of data-measurement tools and storage technology,we can collect complex and intensive data.So how to mine valuable information from massive data has become one of the current researches.Functional data analysis(FDA)is a statistical method for high-dimensional data.Its nature is to treat functional data as in infinite dimensional function space.Nowadays,FDA has an important effect in many fields,such as economics,meteorology,medical diagnosis,and brain images.Classical functional linear regression model is aiming to establish the relationship between the functional covariates and a continuous response variable,which is significant to the information mining.We add random effect to the functional linear regression model,called functional linear mixed effect model,which can capture the individual random effects and the individual random effect slope.However,the addition of random effects leads to the "dimensionality scarcity" of random effect slope.On the other hand,the existing functional linear mixed effect model has some limitations in the application.In this paper,the main work and contribution includes the following three parts.Firstly,the models involved in this paper consider the random effect slope function,which is not considered in most literatures at present.To avoid the "dimensionality disaster",this paper expands the random slope using truncated basis function,which is based on the idea of functional principal component(FPCA).Secondly,this paper proposes the functional generalized linear mixed effect model.We linearize the model by the spline basis function and approximate the profiling function by the first order Laplacian approximation.The simulation takes the binomial distribution and Poisson distribution as examples,which shows the validity and stability of the estimation.When the response variable is normal,the estimation effect is very similar to the result of Liu et al.(2017).Finally,this paper analyzes the effect of multiple sclerosis on the cognitive function based on the DTI data.Comparing to other models,this paper gets a better explanation.Thirdly,this paper proposes the functional semi-parametric mixed effect model,which adds the non-parametric part to the functional linear mixed effect model.Based on the linearization of the model,the parameters and nonparametric parts are estimated by the spline-penalty likelihood method.The Monte Carlo simulations show that the estimated coefficient,the estimated slope function and the estimated nonparametric part are stable and effective.In addition,the models proposed by this paper are suitable for the existence of the functional variables' measurement error by smoothing the covariance matrix of the functional variable.Finally,this paper analyzes the effect of the Ozone pollution on non-accidental mortality based on the NMMAPS data set.