Pure Mathematics
Vol.4 No.06(2014), Article ID:14309,5 pages
DOI:10.12677/PM.2014.46032

Moment Consistency of Kernel Regression Estimation for Random Design

Xin Yang

Department of Mathematics, Guilin University of Aerospace Technology, Guilin

Email: xinyang_emily@sina.cn

Received: Sep. 19th, 2014; revised: Oct. 15th, 2014; accepted: Oct. 20th, 2014

ABSTRACT

For the nonparametric regression model with random design, we discuss the Nadaraya-Watson type kernel regression estimator for mixing samples, and prove the point moment consistency and the global moment consistency of the kernel regression estimator. The obtained results generalize the Devroye’s (1981) conclusions.

Keywords:Random Design, Mixing Sample, Kernel Regression Estimator, Moment Consistency

Email: xinyang_emily@sina.cn

1. 引言

, (1.1)

, (1.2)

2. 主要结论

. (2.1)

. (2.2)

(A1)混合的随机向量序列，且同分布。

(A2) 存在正常数，使得(当时)，其中d为随机向量X的维数。

(A3) 存在正常数和r，使得

. (2.3)

. (2.4)

3. 定理的证明

. (3.1)

. (3.2)

. (3.3)

, (3.4)

. (3.5)

. (3.6)

(3.7)

. (3.8)

(3.9)

. (3.10)

. (3.11)

(3.12)

(3.13)

(3.14)

. (3.15)

. (3.16)

. (3.17)

。对任意的常实数，有

(3.18)

. (3.19)

. (3.20)

(3.21)

. (3.22)

(3.23)

, (3.24)

。显然，，且

(3.25)

(3.26)

(3.27)

. (3.28)

. (3.29)

。由于，所以。因此，函数满足要求。证毕。

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