数学物理学报(英文版) ›› 1985, Vol. 5 ›› Issue (2): 175-185.

• 论文 • 上一篇    下一篇

MEAN SQUARE ERROR FOR UNIFORMKERNEL ESTIMATE AND NEAREST NEIGHBOR ESTIMATE OF NONPARAMETRIC REGRESSION FUNCTIONS

孙东初   

  1. Dept. of Math., East China Normal University, Shanghai, China
  • 收稿日期:1983-12-07 出版日期:1985-06-25 发布日期:1985-06-25

MEAN SQUARE ERROR FOR UNIFORMKERNEL ESTIMATE AND NEAREST NEIGHBOR ESTIMATE OF NONPARAMETRIC REGRESSION FUNCTIONS

Sun Dongchu   

  1. Dept. of Math., East China Normal University, Shanghai, China
  • Received:1983-12-07 Online:1985-06-25 Published:1985-06-25

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摘要： Let (Xi, Yi), i=1, …, n be Rd×R1-valued iid. samples taken from (X, Y). We denote the regression function by m(x)=E(Y|X=x). In this paper we consider the Mean Square Error (MSE) of two usual estimates of m(x0) based on (Xi, Yi), i=1,…, n at a point x0, the Uniform-Kernel Estimate mn(x0) and NN-Estimate mn(x0). We prove that under some reasonable conditions the lowest asymptotic MSE attained for these two estimates have the same form:
C(x0)n-4/(d+4)+o(n-4/(d+4)), where C(x0) is a constant depending on x0. Hence from the point of view of MSE, one has no reason to claim superiority of mn(x0)to mn(x0) or vice versa.

Abstract: Let (Xi, Yi), i=1, …, n be Rd×R1-valued iid. samples taken from (X, Y). We denote the regression function by m(x)=E(Y|X=x). In this paper we consider the Mean Square Error (MSE) of two usual estimates of m(x0) based on (Xi, Yi), i=1,…, n at a point x0, the Uniform-Kernel Estimate mn(x0) and NN-Estimate mn(x0). We prove that under some reasonable conditions the lowest asymptotic MSE attained for these two estimates have the same form:
C(x0)n-4/(d+4)+o(n-4/(d+4)), where C(x0) is a constant depending on x0. Hence from the point of view of MSE, one has no reason to claim superiority of mn(x0)to mn(x0) or vice versa.

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