文档介绍：IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992 1323
Global Convergence of the Recursive Kernel
Regression Estimates with Applications in
Classification and Nonlinear System
Estimation
Adam Krzyzak, Member, IEEE
Abstract-An improved exponential bound on the L, error arbitrary distribution and distribution-free convergence re-
for the recursive kernel regression estimates is derived. It is sults are available, that is convergence can be obtained for all
shown, using the martingale device, that weak, strong and input distributions and for all nonlinear regressions. Distribu-
complete L, consistencies are equivalent. Consequently the con-
tion-free results are the most natural results in the context of
ditions on the smoothing sequence c:=,hf 00 and
> o + nonparametric estimation, since we would like the estimates
~~=lh~~~h.,rJ/~jn=Ih~as n -+ 03 for all e are necessary
and sufficiknt for strong L, consistency of the recursive kernel to extract all statistical information from the observations
regression estimate. The rates of global convergence are also only without presupposing anything about the reality. From
given. Obtained results are applied to recursive classification now on, we focus our attention on random design approach.
rules and to nonlinear time series estimation.
Distribution-free pointwise convergence of the kernel regres-
Index Terms-Nonparametric estimation, recursive kernel re- sion and classification rules has been demonstrated by De-
gression estimate, convergence, rate of convergence, classifica- [8], et al.
tion, nonlinear time series. vroye [9], Krzyzak and Pawlak [27], Greblicki
[17], and Krzyzak [29]. Similar results for the recursive
kernel estimate have been shown by Krzyzak and Pawlak
[26], [28], Gyorfi [20], and Greblicki and Pawlak [19]. The
I. INTRODUCTION
rates of convergence for kernel regression classification rules
EGRESSION function estimation p