Data Sharpening Method Based On Gasser-Muller Estimator

To reduce bias of nonparametric estimator, Choi, Hall and Rousson(2000) suggested methods called data sharpening which based on Nadaraya- Watson estimator and local linear estimator respectively. Data sharpening methods adjust explanatory and/or response variables prior to substitution into Nadaraya-Watson estimator or local linear estimator. It can reduce the bias of the estimator effectively. Data sharpening attempts to retain the advantages, even enhancing them, while adjusting the data so as to overcome the shortcomings, for example, the sensitivity to design sparsity problems. The methods are very effective for low-order nonparametric estimators. To reduce bias, high-order method were used formerly. But they are very sensitive to design sparsity problems. Data sharpening methods produce a reduction in bias and the sensitivity to design sparsity problems.Gasser-Muller estimator is a common kernel estimator. Compared to Nadaraya-Watson estimator, it has a smaller bias. And compared to Nadaraya-Watson estimator and local linear estimator, proofs of theoretical properties of Gasser-Muller estimator are more simple. So this paper suggests data sharpening method based on Gasser-Muller estimator. This paper discusses the properties of data sharpening method based on Gasser-Muller estimator, computes the representation of it and suggest region selection methods so as to enhance the effect. Through the simulation we analyse the advantages and shortcomings of data sharpening method based on Gasser-Muller estimator.This master dissertation consists of five parts. Chapter one introduces the ad- vantage of nonparametric estimator and the significance of data sharpening methods. In Chapter two, some preliminary knowledge about nonparametric estimation is introduced. Section 2.1 introduces three kinds of kernel estimator. Section 2.2 introduces edge effects and how to deal with edge effects. Section 2.3 introduces local linear estimator and its properties. Comparisons between kernel estimators and local linear estimator are also introduced. Section 2.4 provides theoretical properties of Gasser-Muller estimator and the foundation of proof in Chapter five.In chapter three, this paper suggests the data sharpening method based on Gasser-Muller estimator. Section 3.1 introduces data sharpening methods based on Nadaraya-Watson estimator and local linear estimator suggested by Choi, Hall and Rousson(2000) respectively. In section 3.2, we suggest data sharpening method based on Gasser-Muller estimator. We provide the methodology, conclude the advantages and shortcomings between data sharpening methods based on Nadaraya-Watson estimator, local linear estimator and Gasser-Muller estimator. We prove that the bias of data sharpening methods based on Gasser-Muller estimator is O(h~4) which is much better than that of conventional Gasser-Muller estimator. Since not every point in the interval is suitable to apply the data sharpening methods, we suggest two region selection methods for data sharpening. Through the simulation, we conclude that one of them is efficacious.We show the conclusion of simulation in chapter four. Chapter four shows the advantages and shortcomings of data sharpening method suggested in this paper and prove that region selection methods for data sharpening is effective. In chapter five, we prove the bias of data sharpening method based on Gasser-Muller estimator.The signification of this paper is that the bias of Gasser-Muller estimator is reduced effectively by data sharpening method. And the bias is proved to be O(h~4). Furthermore, the region selection methods for data sharpening are suggested and proved to be efficient by simulation.