Adaptive Local Polynomial Regression Models

The traditional local polynomial regression models(LOPO)are widely used due to its simple constructions,fast computations and completely theoretical studies.When the sample points are equally spaced and the error terms are independently and identically distributed,the models have good estimates.However,if conditions axe not satisfied,for example,when the independent variables are non-uniformly distributed,the estimates axe poor.Then the local polynomial regression models with variable bandwidth(LOESS)are developed,whose local weights axe connected with the sparsity of the sample points of independent variables.LOESS considers only the distribution of independent variables,but not the changing characteristics of dependent variables.Particularly,if the data has the heterogeneousness along with the dependent variable,LOESS cannot work well.In this paper,a new adaptive local polynomial regression model(ALOPO)is presented,whose local weights are constructed based on the Manhattan distance which combines the effects form both de.pendent and independent variables.The asymptotic bias,variance of the new method are compared with LOPO.The results shows that the convergence rate of mean squared error of the new method is faster than LOPO under some conditions.Simulations and applications show that ALOPO outperforms LOPO and other methods.Furthermore,the multivariate ALOPO is also studied.