Abstract
A commonly used semiparametric partial linear model is con- sidered. We propose analyzing this model using a difference based approach. The procedure estimates the linear component based on the differences of the observations and then estimates the nonparametric component by ei- ther a kernel or a wavelet thresholding method using the residuals of the linear fit. It is shown that both the estimator of the linear component and the estimator of the nonparametric component asymptotically perform as well as if the other component were known. The estimator of the linear com- ponent is asymptotically efficient and the estimator of the nonparametric component is asymptotically rate optimal. A test for linear combinations of the regression coefficients of the linear component is also developed. Both the estimation and the testing procedures are easily implementable. Nu- merical performance of the procedure is studied using both simulated and real data. In particular, we demonstrate our method in an analysis of an attitude data set as well as a data set from the Framingham Heart Study.
Highlights
Semiparametric models have received considerable attention in statistics and econometrics
By using higherorder differences [34, 35] showed that the bias induced from the presence of the nonparametric component can be essentially eliminated
He constructed an estimator of the linear component and showed it to be asymptotically efficient under the condition that the nonparametric function f is fixed and has a bounded first derivative
Summary
Semiparametric models have received considerable attention in statistics and econometrics. In these models, some of the relations are believed to be of certain parametric form while others are not parameterized. We consider the following semiparametric partial linear model. Where Xi ∈ Rp, Ui ∈ R, β is an unknown vector of parameters, a is the unknown intercept term, f (·) is an unknown function and ǫi’s are independent. Identically distributed random noise with mean 0 and variance σ2 and are independent of (Xi′, Ui)
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