Large Sample Properties Of Gass-Muller Regressioweighted Estimation For Dependent Sequences

The nonparametric regression estimation is a useful tool in researching regression model. It is applied widely in financial economics, such as financial asset price and volatility of return rate. In nonparametric regression estimation, the weighted estimator of regression function is always employed. Since Sotne (1977) proposed the weight function of estimation method which is belonged to nonparametric regression estimates, this method is caused a wide attention. For fixed design regression model Yi=g(xi)+εi,1≤i≤n, Gasser and Muller[1](1979) introduced the weight function is called as Gasser-Miiller type weight function regression estimation, where K(-) is Borel mea-surable function,0<hnâ†'0(when nâ†'∞). Since Gass and Muller put forward the Gasser-Miiller regression weighted right estimate, some scholars have studied it. Roussas G. G., Tran L. T. and Ioannides, D. A.[3](1992) discussed the asymptotic normality under the mixed sequence. But they didn’t provide the convergence rate. Jianqing Fan [4](1992) made the perform simula-tion through two model and changing sample size to compare the mean square error and linear smoothness of the Gasser-Miiller regression right estimate, Nadaraya-Watson regression weighted estimation and the local linear regression estimation. Xiaoling Dou and Shongo Shirahata [5](2009) made some evaluating criteria and some simulations of the programming design with the help of Gasser-Muller regression right evaluation. The simulation results show that the Gasser-Miiller regression weighted estimates is better the local polynomial estimates.However the theory of the Gasser-Muller regression weighted estimation is researched sel-dom. Therefore, to research its theory not only can improve the nonparametric regression theory but also has an important practical significance to research this estimates.In this paper, the author studies the large sample properties of the Gasser-Muller regression weighted estimate for the p mixed sequence and NOD sequence case. And the main research contents and the results are as follows:First of all, discuss the strong consistency of the Gasser-Muller regression right estimate of the strong consistency with the help of in the sample for p mixed sequence and NOD sequence case. extend the corollaries of theorems in literature[13] and weaken the conditions of the theorem in literature[7].Second, discuss the uniformly asymptotic normality of the Gasser-Muller regression weighted estimate for p mixed sequence case. And then provide the uniformly asymptotic normality con-vergence speed. Its convergence rate is about n-1/6.Third, do some numerical simulation for the p mixed sequence and NOD sequence. From the numerical simulation results, the author finds that when the sample size is more, the estimated error is smaller and the accuracy is higher.Finally, the author selects Shanghai Stock180index of Chinese stock market and the Shanghai Stock financial index to do empirical research.