Empirical Euclidean Likelihood Method Of Longitudinal Partially Linear Model With Measurement Error

Longitudinal data is a data model which is common in our daily life, and the data is got by repeatedly observed from the individual in different time or space. It not only overcomes the draw-backs of cross-section data and time series data, but also can effectively considering the advantages and characteristics of the cross-section data and time series data. Engle etc.(1986) put forward a partial linear model to analyze the relationship between the power demand and climate change. Due to the partial linear model combines the characteristics of linear model and nonparametric model, thus it is of more flexibility. In real life, because of the influence of the factors such as mea-suring instruments, the longitudinal data we get may have some measurement error. If we don’t consider the influence of the error, then the final estimate we get will no longer meet unbiasedness, consistency, etc. Therefore, the study of the statistical inference on the longitudinal partial linear model which has measurement error has a strong practical value.In this paper, on the basis of predecessors’research results, we research the estimation of the longitudinal partial linear model which has measurement error with the aid of the empirical Eu-clidean likelihood method. In the first chapter, we introduce the concept of longitudinal data and its research background, then introduce the main problems that we want to study in this paper. In the second chapter we discuss the empirical Euclidean likelihood method in a longitudinal partly linear model, and offer the point estimation of the parameter β and unknown functions θ theo-retically, then construct the confidence interval of estimator, and then prove that the asymptotic distribution of the estimator is chi-square distribution. In the third chapter, we apply the empirical Euclidean likelihood method to discuss the partial linear model which with measurement error, and compute the estimators respectively under the condition that the covariance of the measurement error is known and unknown, and then construct the confidence interval of estimators. At last, nu-merical simulations are conducted to study the problems in this paper, we mainly use the empirical likelihood method and empirical Euclidean likelihood method to compute the coverage and inter- val estimation, then we compare the calculated results of the two methods. From the simulation results, we find that the coverage and interval estimation get from the two methods are close, but in terms of the theoretical calculation, the empirical Euclidean likelihood method is simpler on calculation, it has strong practical application value.The feature of this paper is mainly reflected in the following two aspects:1. In this paper,in order to analyze their asymptotic properties, we apply empirical Euclidean likelihood method into the longitudinal partial linear model which has measurement error. Using empirical Euclidean likelihood method to estimate the parameters can obtained explicit solution which is simpler than other methods.2.The conclusion of this paper can enrich and improve the theory of the linear model of lon-gitudinal data which has measurement error, and has strong theoretical and practical application value.