The Empirical Likelihood Estimation Based On Quasi-Likelihood Function For Partially Linear Models

The present paper is discussed the model with structure: and suppose that where Y is the response variate,(X,T) is the associated covariates, m(·) is an unknown measurable function, e is independent of (X, T), and its mean is zero, β,σ2are unknown parameters, V(·) is a known variance function.In1986, Engle, Rice, Grabger and Weiss were among the first to con-sider the partially linear models when they analyzed the relationship between temperature and electricity usage. Including parametric component and non-parametric component, partially linear models allow easier interpretation of the effect on each variable, so they are more flexible and universal than the classic linear models or non-parametric models, partially linear models attract many statisticians’attentions since the proposed.Likelihood method is a widely used parametric and non-parametric esti-mation method. In this paper, we apply quasi-likelihood method and empirical likelihood method to estimation of partially linear models.Firstly, we structure Nadaraya-Watson estimation mβ of the non-parametric function m(·), and replace m(·) with mβ, we obtain the quasi-score function and the equation of quasi-likelihood estimation of parameter β; Secondly, applying the empirical likelihood method, we use the quasi-score function and variance structure of the model to structure estimating equations, the paper shows that the maximum empirical likelihood estimation of parameter β the same as it-s quasi-likelihood estimation at the moment, and obtain that the asymptotic variance of quasi-likelihood estimation of parameter (βτ,σ). In order to make better use of information on the variance, we make that the variance structure was weighted, and obtain asymptotic variance of empirical likelihood estimation of parameter (βτ,σ), the paper shows that it smaller than asymptotic variance of quasi-likelihood estimation in some conditions. This is to say that we improve the efficiency of parameter estimation of the quasi-likelihood in partially linear models; Finally, we obtain the optimal weight function under certain conditions.