Semiparametric Regression Model For Longitudinal Data

Longitudinal data is referred to data in which the same sample or individual is measured repeatedly through time or space, so it combines elements of cross-sectional data and time-series data. It combines the characteristics of cross-sectional data and time-series data. It is observed and got in the order of time by the same sample, so longitudinal data can analyze effectively the change of individuals and variation among individuals. It plays an important part in applications contrast to cross-sectional data or time-series data.The main methods in studying longitudinal data are parametric method and, non-parametric method and semiparametric method. The main content of this paper is as follows:(1) Based on the rational analysis of differences in natural among cross-sectional data and time-series data. The paper analyses and summarizes the parametric method ofstudying longitudinal data------Quasi-likelihood Estimate, Generalized Estimate Equationand quasi-least Square, and points out their finites. It focuses on the optimality of the quasi-least square estimating equations, and give out its iterative algorithm.(2) This paper use least square kernel to estimate the unknown coefficients of semiparametric linear regression model for longitudinal data, for the difficult in dealing with thedata' correlation and theoretically computes out the estimates of β and g(T).(3) This paper studies the coefficients' estimates in semiparametric time-varying coefficients regression model for longitudinal data by the numbers. Firstly we use the weighted least squares to type estimators for the unknown parameters of the parametric coefficient functions as well as estimators for the nonparametric coefficient functions are developed. Secondly it proves that kernel neighborhood smoothing is asymptotically more efficient than a single nearest neighbor smoothing. Finally the asymptotic optimal bandwidth is derived, and Location-Shift invariantproperty of the estimates of β and g(T) is also proved.(4) Demonstrations show that least square kernel is a wonderful way to decompose the correlations of semiparametric linear regression model for longitudinal data, and weighted least square kernel smoothing and weighted least square are viable and effective. It shows that the kernel neighborhood smoothing is asymptotically more efficient than a single nearest neighbor smoothing in some cases when the number of observations per subject is small, whereas the difference will decreases as the number of observations per subject increase.