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| The Strong Consistency of Error Probability Estimates in NN Discrimination |
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Citation： |
Bai Zhidong.The Strong Consistency of Error Probability Estimates in NN Discrimination[J].Chinese Annals of Mathematics B,1985,6(3):299~308 |
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Net amount： 739 |
Authors： |
Bai Zhidong; |
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| Abstract： |
Let(X, \theta), (X_1,\theta_1),\cdots,(X_n,\theta_n) be iid.R^d\times{1, 2,\cdots,s}-valued random vectors and let L_n be the posterior error probability in NN (nearest neighbor),discrimination. Some knowledge of the unknown value of L_n is of great meaning in many applications. For this aim, in 1971, T. J. Wagner introduced an estimate of L_n which is defined by
$[\tilde R]_n=1/n[\sum\limits_{j = 1}^n {I({\theta _j} \ne {\theta _{nj}})} \],$
where \theta_nj is the NN discrimination of \theta_j based on the training samples (X_1,\theta_1),\cdots,(X_j-1,\theta_j-1),(X_j+1,\theta_j+1),\cdots,(X_n,\theta_n).Then he showed that $[{\hat R_n}\xrightarrow{P}R\]$, where R is the limit of the prior error probability. But the problem of" [{\hat R_n}\xrightarrow{a.s}R\]” is still left open since that time. In this paper, it is shown that for any \varepsilon>0, there exist two positive constants a and C such that P(|[\hat R_n]-R|\geq \varepsilon) \leq Ce^-an. By this it is clear that [{\hat R_n}\xrightarrow{a.s}R\]. |
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