Parametric Estimation Of Extended Partially Linear Single Index Model

Partial linear single index model(PLSIM)is a semiparametric regression model which is widely used in many fields.However,in the previous parameter estimation methods of this model,most of the estimators need to solve optimization problems with complicated computation.Based on this,this paper designs a noniterative and easily computed estimation method for discrete variables in PLSIM.this paper proposes an extended partial linear single index model(EPLSIM),which is applicable to both multidimensional continuous variables and discrete variables,and focuses on the estimation of parameters corresponding to continuous variables and discrete variables in EPLSIM.In this paper,we first start with the partial linear single index model and introduce discrete variables into the model to establish an extended partial linear single index model.Then,we make three assumptions about the parameter space,nonparametric link function and support of variables in EPLSIM.Finally,the theorem of EPLSIM's identification is deduced according to the assumed conditions and illustrates the uniqueness of the parameters in EPLSIM.Thus,the preconditions for constructing parameter estimators of the extended linear single index model are obtained.Secondly,based on EPLSIM and its identification theorem,this paper focuses on parameter estimation in the model,and constructs estimators for parameters corresponding to continuous variables and discrete variables.The local polynomial regression and outer product gradient estimation method(OPG)are used to construct the parameter estimators corresponding to the continuous variables.The boundary effect problem is solved by defining the trimming function,and the iterative optimization of estimators is carried out by using the adaptive kernel function instead of the ordinary kernel function.For the construction of parameter estimators corresponding to discrete variables,this paper designs a noniterative and easily computed parameter estimation method,which does not need to solve the optimization problems with complicated computation.Finally,Monte Calro experiment is carried out on the constructed parameter estimators for two different EPLSIM and the auto data is applied in EPLSIM.In the experimental algorithm,the adaptive kernel function is used to iteratively optimize the estimators,and Gauss-Legendre quadrature with degree 5 is used to calculate the estimators,so as to obtain the parameter estimators corresponding to continuous variables and discrete variables.By analyzing and comparing the mean and standard deviation of the parameter estimators produced in the experiment,it can be concluded that the estimators based on the extended partial linear single index model are effective.Meanwhile,based on the application of auto data by EPLSIM,the practicability of EPLSIM is illustrated by analyzing the error rate between the predicted value and the real value.