Variable Bandwidth M-estimations Of Partial Linear Model

The partial linear regression model was provided by Engel etc when they studied thatthe weather influenced on power demand. The partial linear regression model containsboth the parametric part and the nonparametric part, and it is more flexible than linearmodel freedom. It attracts many scholars to study the consistency and asymptoticnormality behavior of the parametric part and the nonparametric part in the partial linearregression model. There have been many estimation methods for studying the partiallinear regression model, such as nuclear estimation, spline estimation, dividing polynomialestimation and so on. However, these regression methods are based on least squaremethod and lack of robustness. In this dissertation, M-method in the robust statistics isused to discuss the estimations of the unknown function and the unknown parameter of thepartial linear regression model with function relation and the partial linear regressionmodel. The main contents are as follows:First, the dissertation has introduced the development history and research status ofmodel, and the knowledge related with M-estimations.Second, the dissertation has built the variable bandwidth M-estimations of theunknown function and the unknown parameter of the partial linear regression model withfunction relation by M-method and embedding variable bandwidth, and has proved theconsistency and asymptotic normality of M-estimations. Further it has discussed variablebandwidth one-step local M-estimations, and one-step local M-estimations have theasymptotic normality the same as M-estimations when beginning estimations are goodenough. One-step local M-estimations not only steadies local least square estimation andreally inherits all good nature of local least-square estimation, but also reduces thecomputational complexity.Third, the dissertation has discussed the existence and uniqueness of variablebandwidth M-estimations of the unknown function and the unknown parameter in thepartial linear regression model.Finally, the dissertation has combined weighted least squares estimation with M-estimation, and has discussed convergence speed and strong consistency of theunknown function and unknown parameter estimation in the special partial linearregression model.