The Generalization Of Double-smoothing Local Linear Regression In Bivariate Data

Local linear regression is a particular situation of local polynomial regression. It has excellent theoretical and intuitive properties. The local linear estimator can achieve full minimax efficiency, and the qualities of asymptotic bias and varianc are superior. However, in practice, the local linear estimator have more rough appearance when the distribution of the design points is more sparse. This problem is caused by the essence of the local linear fit.Hua He and Li-shan Huang(2009) propose a new estimator,the double-smoothing local linear estimator. This new estimator improve the conventional local linear estimator by weighted integrally combining conventional local linear estimator and its derivative function. This estimator tries to make full use of data information in x’s neighborhood. Under the condition of without changing the order of variance, the double-smoothing estimator reduces the order of bias from h2 to h4. Because of the design matrix was only used in first step of smoothing, the new estimator has less variability to overcome the sparse data problem.In this paper, we expand the double-smoothing local linear estimator into bivari-ate data set. We derive the explicit representation of asymptotic bias and asymptotic variance. The results can provide some help to obtain the smoothing parameters.