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| ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES |
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Citation： |
Chen Xiru.ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES[J].Chinese Annals of Mathematics B,1984,5(2):185~192 |
| Page view： 676
Net amount： 801 |
Authors： |
Chen Xiru; |
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| Abstract： |
Let $\[{X_1}, \cdots ,{X_n}\]$ be iid samples drawn from an m-dimensional population with a probability
density f, belonging to the family $\[{G_k}\]$ i.e. the family of all densities whose partial derivatives of order k are bounded by $\[\alpha \]$. It is desired to estimate the value of $\[f\]$ at some predetermined point $\[\alpha \]$, for example $\[\alpha = 0\]$. Farrell obtained some results concerning the best possible convergence rates for all estimator sequence, from which it follows,for example, that there exists no estimator sequence $\[\{ {\gamma _n}(0) = {\gamma _n}({X_1}, \cdots ,{X_n},0)\} \]$ such that $\[\mathop {\sup }\limits_{f \in {C_{k,\alpha }}} {E_f}{[{\gamma _n}(0) - f(0)]^2} = o({n^{ - 2k/(2k + m)}})\]$. This article pursues this problem further and proves that there exists no estimator sequence $\[\{ {\gamma _n}(0)\} \]$ such that
$$\[{n^{ - k/(2k + m)}}({\gamma _n}(0) - f(0))\mathop \to \limits^{{P_f}} 0,for\begin{array}{*{20}{c}}
{each}&{f \in {C_{k,\alpha }}}
\end{array}\]$$
where $\[\mathop \to \limits^{{P_f}} \]$ denotes convergence in probability. |
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