Estimators For The Partially Linear Models

The present paper is discussed the model with structure:where Y is the response variable, (X, Z) is the associated covariates.εis the random error, it is independent of(X, Z) with E(ε) = 0, Var(ε) =σ~2. g(·) andβare respectively a unknown smooth functions and p _dimensional vector by unknown parameters.The model has been proposed and applied by Engel, etc. to study the effect of weather on electricity demand. This model is much more flexible than the standard linear model since it combines both parametric and nonparametric components. The increasing recognition of partially linear models has attracted a number of authors to study the asymptotic behavior of both the parameter and function estimates.There are many methods for partially linear models: kernel, spline, ployno-minals, etc. But all the mentioned references are based on the least squares facilitates computation. The M-type objective function is often used for robust estimation.Firstly, this paper focuses on establishing joint asymptotic normality of the partially linear models M-type estimators of regression function and its associat- ed deriavative, based on the local linear regression smoothers implemented with variable bandwidth. Parametric component on partly linear models is developed by average method. Consistency of the estimators are investigated.Secondly, nonparametric component on partially linear models is developed by one-step local M-regression methods, but which is enhanced via incorporating a variable bandwidth scheme. This allows the resulting estimation procedure to cope well with spatially inhomogeneous curves, heteroscedastic errors and highly nonuniform designs. Then one-step local M-estimators are shown to have the same asymptotic behaviors as their corresponding M-estimators. Parametric component on partially linear models is developed by average method. Consistency of the estimators are investigated.It is necessary to mode and analyze longitudinal data in statistics, that are common complicated data in biostatistics, medicine and economics. Partially linear models are very important and useful in order to reduce the curse of dimensionality in statistical inferences. Finally, we study partially linear models for general longitudinal data. The profile least-square estimators for the parametric component and nonparametric component are proposed by local linear regression techniques. Consistency and asymptotic normality of the proposal estimators are established.