The Empirical Likelihood Ratio Test For The Partial Linear Model

The partial linear model is firstly put forward by Englc and so on.,when they studied the relation between weather and electricity sales in1986. This mod-el contains both the linear parametric part and the non-parametric part, is more practical than the linear model and more comprehensive than the non-parametric model,so it was widely used.First, we introduce the source of a partially linear model, as well as the research status. At present, the estimation method of a partially linear model has matured. However, model test is still not very mature. So, in the next section we introduce some tests raised by the statistical experts and the limitations of those tests, mainly contained the test based on the likelihood function, based on the residual-marked process, and based on the adaptive Neyman test.The research model in this article is where X is the d-dimensional random vector, T is d1-dimensional random vector, β is d-dimensional unknown parameters, g(·) is a smooth function. When X, T are given,the expectation of ε is zero,the variance is σ2. The hypothesis isIn this article, we use the empirical likelihood ratio (ELR) statistic which was first put forward by Owen, to give the test. In section2.1, we give the details of the ELR statistic.We can get the data (t1,x1,y1),(t2,x2,y2),…,(tn,xn, yn) throw model (1.3). Then we define the φi, x is the mean of the samplie.Then,give the ELR statistic of the hypothesis, φi is the estimated value of φi. We can find out the estimated value of λH0by Lagrange multiplier method and the weighted least squares estimate method. This value converges in distribution to χ2distribution. This test has many advantages, for example it doesn’t involve any variance estimations, doesn’t constraint the range of the confidence regions, and is the Batlett rectification and so on.