Statistical Inference And Application For Linear Mixed-effects State Space Model

State space models, which are based on the theory of recursive Bayesian filtering and modern statistical method, provide a consistent analytical framework for the extensive time series analysis. State space models are widely used in the natural and social sciences. Longitudinal data, which can reflect the individual difference and the general trend of the population, is a fusion of time series data and panel data. It is widely distributed in economics, medicine and biology. Mixed effects model are a powerful tool to analyze longitudinal data. It has attracted much attention in recent years that the combination of mixed effects model and state space models are applied to deal with the longitudinal data. Thus it is practical significance to further study the statistical inference of MESSM.For parameter estimation in MESSM, three algorithms are discussed based on the maximum likelihood estimation, containing the weighted grid points approach, the EM algorithm and the Newton-Raphson method. The EM method is easy to be carried out and global convergence. It is advised to find the estimating initial value because of its slow convergence speed. For the EM algorithm which is based on disturbance smoothing, the explicit formulations are given for the autoregressive plus noise model and local linear model respectively. As for the Newton-Raphson method, the MESSM scored vector is induced easily under the normality assumption, from which it is shown that the maximum likelihood estimation is equivalent to the moment estimation and its convergence is better than EM method.For the state estimation of MESSM, without assuming the random effects known in advance three different methods are investigated, that are the combination of mixed Kalman filtering and sequential Monte Carlo sampling, metropolis move and kernel smoothing respectively,the individual random effects and state are estimated at the same time.Last the simulation studies are carried out based on the simulated data from autoregressive model, the results show that MKF-MM and MKF-KS have the most minimal MSE, and MKF-KS has the shortest operating time.Besides, two-stage MESSM is applied to real clinical data, MKF-KS is applied to estimate state, the results show that the proposed algorithms have better performance than the existing methods.