Local Least Absolute Relative Error Estimating Approach for Partially Linear Multiplicative Model

The partially linear multiplicative regression model is considered. This model, which becomes a partially linear regression model after taking logarithmic transformation, is useful in analyzing data with positive responses. Chen et al. (2010) mentioned that in many applications the size of relative error, rather than that of error itself, is the central concern of practitioners. We extend the criterion of least absolute relative error (LARE) to the partially linear multiplicative regression model by local smoothing techniques. Consistency and asymptotic normality are investigated. We utilize a random weighting method to estimate asymptotic covariance of the parameter estimator. We also propose a simple and effective method to select important variables in the linear part. The oracle property (Fan and Li (2001)) is proved. Some numerical studies are conducted to evaluate and compare the performance of the proposed estimators. The body fat dataset is analyzed for illustration.