The Related Estimates And Properties Of Adaptive Varying-coefficient Models With Data Missing At Random

An adaptive varying-coefficient model is considered when the responses are missing at random (MAR). In practice, some parts of the explored matrix are often not responsive. Due to the high costs, some data of the design can't be completed. In medical statistics, there are many cases which would lead to loss data. For example, the patients participated in the study had died or have been transferred, medical records are missed and so on. In order to reduce the impact of missing data, we can deal with incomplete data and make full use of statistical data. We consider the adaptive varying-coefficient model: where be a random sample of incomplete data from this model, whereδi=1 if the yi is observed andδi=0 otherwise,i= 1,2,…,n, Unknown quantity are called the coefficient functions, {εi}i=1n are errors with independent identical distributions and E(εi)＝0,Var(εi)＝σ2.Throughout this paper, we assume that y is missing at random (MAR). It implies that 8 and y are conditionally independent for x given, Furthermore, in the context of the response variable missing at random, the following three problems are discussed:(1) The local estimation with the complete data. The local linear method is employed to give the estimation of coefficient functions gi(·) with givenβand the asymptotical normality of all estimations is given. And then, using one step iterative estimation to discuss the estimationsβwith gi(·)s fixed. Using back-fitting algorithm to discuss the estimations gi(·) andβwhen they are unknown.(2) The locally weighed linear estimation. Using the inverse selection probability as the weighted, the local linear method is also employed to give the estimation of coefficient functions gi(·) with givenβand the asymptotical normality of all estimations is given. And then, using one step iterative estimation to discuss the estimationsβwith gi(·)s fixed. Using back-fitting algorithm to discuss the estimations gi(·)andβwhen they are unknown.(3) The local regression estimation with the imputed values. The imputation method consists of two steps. The first step involves imputing missing response values; the second step, we substitute yi* byThen, the same estimation techniques based on imputed are applied to obtain the efficient estimations of gi(·) with givenβand the asymptotical normality of all estimations is given. And then, using one step iterative estimation to discuss the estimationsβwith gi(·)s fixed. Using back-fitting algorithm to discuss the estimations gi(·)andβwhen they are unknown.At last, simulation study of our estimations gi(·)with givenβandβwith gi(·)s fixed by using Matlab is showed and we find the last estimation is better than the others.