Variable Selection For Partial Linear Single–Index Additive Model Based On Quantile Regression

In recent years,research on semi-parametric model has become increasingly popular,it not only has the flexibility of nonparametric model,but also maintains the characteristic of parametric model.Partial linear additive model(PLAM)is an important form of semi-parametric model,so the study of PLAM is also particularly important for the semi-parametric model.In addition,quantile regression has taken an import role to study the conditional distribution of the response variable.Variable selection aims to get the significant variable.In this article,we focus on the study of quantile regression and variable selection.Therefore,this paper proposes applying the quantile regression and variable selection to the PLAM to analyze the medical cost data with complex situation.The study of medical cost data has always been the focus of the health care industry research,and its research is also an important measure to protect the social health care system.In practice,medical cost data often are incomplete and they have extremely complex relevance so that it makes some difficulties in researching.The medical cost data is often skewed,heteroscedastic,non-normal and so on.At the same time because of the existence of binary variables in some explanatory variables,the collected medical cost data have some sparseness,so the traditional models and methods are no longer fit the data.Therefore,this paper proposes a new model which is more flexible to deal with medical cost data.Then we can analyze the heterogeneity and sparseness of the medical cost data by combining the quantile regression and the variable selection method.Research methods of this article Mainly divided into two parts,organized as follows:The first part,we propose minimizing the average quantile loss estimation to realize the quantile regression of PLAM.Firstly,we analysis the quantile regression based on the kernel function.Then the estimation of the parameters and the nonparametric functions are obtained by means of local polynomial regression,which can reach the optimal convergence rate and establish the asymptotic normality.The second part,in the framework of quantile regression,we propose using the adaptive LASSO as variable selection method for the PLAM.The algorithm uses the least angel regression(LARS).We show that the adaptive lasso enjoys the Oracle properties;namely,it performs as well as if the true underlying model were given in advance.Finally,Simulation studies demonstrate that the proposed inference procedure performs well.In addition,this paper shows the corresponding numerical simulation for the above two methods,and proves the asymptotic properties.Through numerical simulation,we found that the estimation methods given in this paper have some advantages for dealing with heteroscedastic and sparseness covariate data,which can get a robust estimate effectively.And the calculation of the second method is more convenient,the operation speed is faster.Furthermore,we use the partial linear additive model to analyze the patients with chronic heart failure(CHF),which records the patient's condition at hospital for more than 60 years of age,The results showed that the non-diseased service situation was significant at all levels,and the race factor was also significantly affected by the improvement of the sub-level.The age and death factors were at a certain level have an impact effect,while gender and tracking time have no effect on medical cost at any quantile level.In conclusion,using the PLAM proposed in this paper and the method of quantile regression and variable selection,we can effectively analyze the medical cost data.