The Estimates And Properties Of Adaptive Variable Coefficients EV Model

The varying coefficient models are introduced by Cleveland, Grosse and Shyu (1991) to extend the application of local regression techniques from one-dimensional to multi-dimensional setting. Fan,J. , Yao, Q.W. and Cai, Z.W.(2000)[1] have proposed the adaptive varying-coefficient models and discussed its properties. In practice, the models have already been widely used in areas such as biology, medical science, economics, finance, and so on.The Errors-in-Variables (EV) models, also called Measurement Error (ME) models, are the regression model in which both dependent and independent variables carry errors. People have studied the EV models for a long time. As early as the late 1900s, people had studied these models (Adcock, 1877, 1878; Kummel, 1879). Fuller (1987) had discussed linear EV models in his monograph 'Measurement Error Model'. Due to its special structure, we need to consider measurement error when dealing with EV models and thus it is more difficult to research than classical regression models. For example, it is hard to explore the existence and the consistence of parameters estimators.In practice, it is inevitable that errors might occur in the observation of independent and dependent variables (e.g. errors that is caused by measurement device). Sometimes, we cannot ignore such errors and thus we propose a new model-Adaptive Variable Coefficient Errors-in-Variablesmodel: where x_i =(?)(x_i,y_i) are random variables in R^p+1×R¹ which are cannot observed accurately and their observations are (X_i,Y_i),(i=0,1,…,n). g_j(·)(j = 0,1,…,p) are bounded continuous functions with g_j (·)≠0 (j = 0,1,…,p).Let u =β^Tx., and (ε_i,e_i^T)^T are p+2 dimensional independent andidentical distributed random vectors with: x_i and e_i, y_i andε_i, x_i andε_i are unrelated and all observations are independent.The study of adaptive varying coefficient models is just the beginning. Though Fan, J. discussed how to estimate its unknown coefficients and parameters, how to choose the bandwidth as well as its applications [1], the theoretical results about this models are rare.The innovation of this paper is that has joined the observation error in the existing adaptive variable coefficient model.In this paper, we applied kernel smoothing method and generalized least squares to the estimation of coefficients of adaptive varying-coeffiient EV models. First, we assume that the coefficients take their mathematical expectations, and thus the model is turned into normal linear models. We use the least squares to get the one-step estimation of the coefficients. Second, substituting the kernel estimators for the coefficients and using the generalized Least Squares, we get the two step estimators of the coeffients.We use the One-step iterative estimation method to get the estimation ofβ. Under some regularity conditions, we get the strong consistency and uniform strong consistency of estimators of coefficient functions .Finally, we use Mat-lab to Simulate the estimations which we have got. According to the results, we conclude that our methods are good.