Empirical Likelihood For Parameters Based On Semi-parametric Regression Model

Empirical likelihood(EL),proposed by Owen(1988),is an influential computation-intensive statistical approach.This method defines an EL ratio function and uses its maximum subject to constraints on the parameters to construct confidence intervals/regions.As a nonparametric likelihood method in the sense of being distributional assumption free,the EL has many desirable advantages in deriving confidence regions for unknown parameters,such as EL-based inference does not involve variance estimation,the shape and orientation of confidence regions based EL are determined entirely by the data and many others.Therefore,EL method has attracted the interest of many statisticians,and they apply this method to a variety of statistical models and various fields.The main content of this paper is to study the EL for model parameters based on semi-parameter regression.Firstly,we study the EL for high-dimensional partially linear model with martingale difference errors.The error is assumed to be dependent,i.e.error is a sequence of martingale difference.we define the corresponding EL ratio(ELR)function..Then we give out some basic conditions and some lemmas to prove the asymptotic property of ELR for this situation and for a linear combination of unknown parameter.Subsequently we carry out a simulation based MATLAB,and shows that the EL method outperforms than the profile least squares.Secondly,we study the EL for partially functional linear model with martingale difference errors.The approximate expressions of partially functional linear model is given by using Mercer's theorem and Karhunen-Loeve expression.Then,we get the ELR and it's asymptotic property.Then,giving some basic conditions and some lemmas to prove this property.Finally,we consider the EL for high-dimensional partially functional linear model.The ELR is shown to have asymptotic normality distribution.Then,giving several basic conditions and some lemmas to prove this property.In addition,a simulation is carried out based MATLAB to show that the EL method performs better than the profile least squares.