关于最近邻回归估计的收敛速度

ON THE CONVERGENCE RATE OF THE NEAREST NEIGHBER REGRESSION FUNCTION ESTIMATOR

摘要: 设（X,Y）,（X₁,Y₁）,（X₂,Y₂）,…为取位于 R^d×R¹上的 iid.随机向量序列,E|Y|<∞.本文研究了回归函数 m（x）的最近邻估计 m_n（x）的强收敛速度问题,在一定条件下证明了它满足重对数律,即\limsup _n \rightarrow \infty \frac\left|m_n(x)-m^i \cdot c\right| i\left(2 \sum_j=1^k \nu_m^2 \log \log n\right)^1 / 2 \leqslant(2 \operatornamevar(Y \mid X=x))^1 / 2 a. в.

Abstract: Let（X,Y）,（X₁,Y₁）,（X₂,Y₂）,…be a sequence of iid.R^d×R¹-valued random vectors with E|Y|<∞.This paper investigats the pointwise strong convergence rate of the nearest neighbor regression function estimator and its iterated logarithm-type rate is obtained,i. e,,under certain conditions \limsup _n \rightarrow \infty \frac\left|m_n(x)-m^i \cdot c\right| i\left(2 \sum_j=1^k \nu_m^2 \log \log n\right)^1 / 2 \leqslant(2 \operatornamevar(Y \mid X=x))^1 / 2 a. в.