Some Research On Estimation Of Coefficient Function With Data Missing At Random

Varying coefficient model is a generalization of linear model and a powerful tool for studying high-dimensional data analysis.Because its coefficient is a function of some factors,it can not only avoid the"Curse of dimensionality" but also greatly reduce the deviation when modeling with varying coefficient model.The varying coefficient model not only retains the interpretability of the parametric model,but also the flexibility and robustness of the nonparametric model.In studying of varying coefficient model,the most popular method is least squares(LS)regression,,combined with local linear approximation of the coefficient function.It is worth noting that when the dimensional difference between variables is too large,the effect of least squares(LS)is often unsatisfactory.Naturally,statisticians take the "relative error criterion" into account.For example,Chen[17]and Chen[18]put forward "least absolute relative error criterion(lare)" and " least product relative error criterion(LPRE)" based on two relative errors.These two criteria are scale invariant,which opens up a new way for the research of regression analysis.At the same time,it should be noted that almost all classical statistical theories are based on complete data sets,but in practice,there are often cases of missing data,which makes statisticians have to establish new statistical methods under the missing data.Considering that one of the factors affects the development of the relative error criterion is the inability to ensure that the response variable is always positive,this thesis naturally selects the varying coefficient product regression model as the research object.When the response variable is failure time,the model is also called " accelerated failure time model",which has been widely used in survival analysis.The main content of this paper is to give the leasst product relative error criterion(LPRE)estimation of varying coefficient product regression model under the random missing of response variables(MAR).In this paper,the idea of Horvitz Thompson inverse probability weighting is used to deal with the missing of response variables,the local binomial is used to approximate the coefficient function,and combined with the least product relative error(LPRE)criterion,a local LPRE estimation based on H-T inverse probability weighting is proposed,and the asymptotic properties of the estimation are studied.Finally,we conduct numerical simulation and draw a conclusion:the estimation obtained by LPRE criterion is more reliable than that by LS criterion;With the increase of sample size,the effects of both estimates are getting better;LPRE criterion combined with local binomial approximation is more reliable than local linear approximation.