Statistical Inference For Generalized Partial Linear Single Index Models With Missing Covariate Data

The Generalized Partial Linear Single Indicator Model(GPLSIM)is widely used in biomedical and socioeconomic research fields.Semiparametric models can avoid the shortcomings of high-dimensional data problems,and take into account the advantages of flexible non-parametric models.The estimators of the models have good statistical properties.In addition,from the situation of obtaining real data,incomplete data of some experiments or observations are common.Therefore,based on the generalized partial linear model,this paper considers the absence of covariate data,and infers the model statistically.In order to solve the problem of missing covariates in the model,the inverse probability weighting method is usually used.At the same time,empirical likelihood method has a good performance in inferring semi-parametric model.Therefore,this paper attempts to combine the two methods as the statistical inference method needed in this paper.The specific contents of this paper are as follows:Firstly,in the first two chapters,the types of missing data,the missing mechanism,the relevant literature on missing data at home and abroad,the processing methods of missing data,the theory of inverse probability weighting method,empirical likelihood and the principle of estimation equation are elaborated in detail.Secondly,in the third chapter,empirical likelihood weighted estimators of unknown parameters of GPLSIM with missing covariate data are constructed by combining empirical likelihood and inverse probability weighted statistical methods.It is further verified that empirical likelihood ratio statistics gradually obey chi-square distribution.In the fourth chapter of this paper,we simulate the data based on R software.We use Sine-bump model to simulate the effect of the two estimation methods to get the corresponding estimators of unknown parameters.Under different missing probabilities and sample sizes,we get the deviation,standard deviation and mean square error of the unknown parameters estimators in the model.The simulation results are compared with the real situation of the model,and the following conclusions are obtained.Firstly,when the missing probability of the model remains unchanged and the number of samples increases,the bias and mean square error of estimators calculated by IPW and IEL decrease;secondly,the sample size is determined,if the missing proportion of the model is increased,then the bias and mean square error of estimators are increased by IPW and IEL;thirdly,if the missing proportion and sample size are determined,the estimation error and mean square error are increased according to the sample size.The absolute deviation,standard deviation and mean square error of the estimator obtained by IEL are less than those of IPW,which shows that the method proposed in this paper is effective.