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| STRONG UNIFORM CONSISTENCY FOR DENSITY ESTIMATOR FROM RANDOMLY CENSORED DATA |
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Citation： |
Zheng Zukang.STRONG UNIFORM CONSISTENCY FOR DENSITY ESTIMATOR FROM RANDOMLY CENSORED DATA[J].Chinese Annals of Mathematics B,1988,9(2):167~175 |
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Net amount： 757 |
Authors： |
Zheng Zukang; |
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| Abstract： |
Let $\[{X_1}, \cdots ,{X_n}\]$ be a sequence of independent identically distributed random variables with distribution function $F$ and density function $f$. The $\[{X_i}\]$ are censored on the right by $\[{Y_i}\]$,where the $\[{Y_i}\]$ are i. i. d. r. y. s with distribution function $G$ and also independent of the $\[{X_i}\]$. One only observes
$$\[{Z_i} = \min ({X_i},{Y_i})\begin{array}{*{20}{c}}
{}&{{\delta _i} = {I_{({X_i} \le {Y_i})}}}
\end{array}\]$$
Let $\[S = 1 - F\]$ be survival function and $S$ be the Kaplan-Meier $\[{\rm{estimato}}{{\rm{r}}^{[3]}}\]$, i.e.,
$$\[S(x) = \left\{ {\begin{array}{*{20}{c}}
{\prod\limits_{{Z_{(i)}} \le x} {{{(1 - \frac{1}{{n - i + 1}})}^{{\delta _{(i)}}}}} ,x < \max {Z_i}}\{0,\begin{array}{*{20}{c}}
{}&{x \ge }
\end{array}\max {Z_i},}
\end{array}} \right.\]$$
where $\[{Z_{(i)}}\]$ are the order statistics of $\[{Z_i}\]$ and $\[{\delta _{(i)}}\]$ are the eorresponping censoring indicator functions. Define the density estimator of $\[{X_i}\]$ by
$$\[f_n^*(x) = \frac{{F(x + {h_n}/2) - F(x - {h_n}/2)}}{{{h_n}}}\]$$
where $\[F = 1 - S\]$ and $\[{h_n}( > 0) \downarrow 0\]$.
In this paper the author uses the strong approximations to get the strong uniform consistency of $\[f_n^*(x)\]$ under certain assumptions and also obtains better
order, i.e.,
$$\[\mathop {\sup }\limits_{ - \infty < x < {T^*}} \left| {f_n^*(x) - f(x)} \right| = O({(\frac{{\log n}}{n})^{2/5}})\]$$
where $\[{T^*} < T = \inf \{ x:H(x) = 1\} \]$ and $\[H(x) = 1 - (1 - F(x))(1 - G(x))\]$. |
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