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| UNIFORM STRONG CONVERGENCE RATE OF NEAREST NEIGHBOR DENSITY ESTIMATION |
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Citation： |
Yang Zhenhai,Zhao Lincheng.UNIFORM STRONG CONVERGENCE RATE OF NEAREST NEIGHBOR DENSITY ESTIMATION[J].Chinese Annals of Mathematics B,1984,5(3):325~332 |
| Page view： 655
Net amount： 889 |
Authors： |
Yang Zhenhai; Zhao Lincheng |
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| Abstract： |
Based on [3] and [4], the authors study strong convergence rate of the $\[{k_n} - NN\]$ density estimate $\[{f_n}(x)\]$ of the population density $\[f(x)\]$, proposed in[l]. $\[f(x) > 0\]$ and $\[f\]$ satisfies $\[\lambda \]$-condition at $\[x(0 < \lambda \le 2)\]$, then for properly chosen $\[{k_n}\]$
$$\[\mathop {\lim \sup }\limits_{n \to \infty } {(\frac{n}{{\log n}})^{\lambda /(1 + 2\lambda )}}\left| {{f_n}(x) - f(x)} \right| \le C\begin{array}{*{20}{c}}
{a.s}&{}
\end{array}\]$$
If $\[f\]$ satisfies $\[\lambda \]$-condition, then for propeoly chosen $\[{k_n}\]$
$$\[\mathop {\lim \sup }\limits_{n \to \infty } {(\frac{n}{{\log n}})^{\lambda /(1 + 3\lambda )}}\mathop {\sup }\limits_x \left| {{f_n}(x) - f(x)} \right| \le C\begin{array}{*{20}{c}}
{a.s}&{}
\end{array}\]$$
where C is a constant. An order to which the convergence rate of $\[\left| {{f_n}(x) - f(x)} \right|\]$ and
$\[\mathop {\sup }\limits_x \left| {{f_n}(x) - f(x)} \right|\]$ cannot reach is also proposed. |
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