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| THE OPTIMAL RATE OF CONVERGENCE OF ERROR FOR kNN MEDIANREGRESSION ESTIMATES |
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Citation： |
Chen Xiru,Zhao Lincheng.THE OPTIMAL RATE OF CONVERGENCE OF ERROR FOR kNN MEDIANREGRESSION ESTIMATES[J].Chinese Annals of Mathematics B,1986,7(2):129~138 |
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Authors： |
Chen Xiru; Zhao Lincheng |
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| Abstract： |
Let $\[(X,Y),({X_1},{Y_1}), \cdots ,({X_n},{Y_n})\]$ be iid. random vectors, where $\[Y\]$ is one-dimensional. It is desired to estimate the conditional median $\[\xi (X)\]$ of $\[Y\]$, by use of $\[{Z_n} = \{ ({X_i},{Y_i}),i = 1, \cdots ,n\} \]$ and $\[X\]$. Denote by $\[{\xi _{nk}}(X,{Z_n})\]$ the $\[\xi NN\]$ estimate of $\[\xi (X)\]$, and put
$\[{H_{nk}}({Z_n}) = E\{ |{\xi _{nk}}(X,{Z_n}) - \xi (X)||{Z_n}\} \]$, the conditional mean absolute error. This artical establishes the optimal convergence rate of $\[P({H_{n{k_n}}}({Z_n}) \ge \varepsilon )\]$, under fairly general assumptions on $\[(X,Y)\]$ and $\[{k_n}\]$, which tends to $\[\infty \]$ in some suitable way. |
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