STRONG UNIFORM CONSISTENCY FOR DENSITY ESTIMATOR FROM RANDOMLY CENSORED DATA

Let X_1,…,X_n be a sequence of independent identically distributed random variableswith distribution function F and density function f.The X_are censored on the right byY_i,where the Y_i are i.i.d.r.v.s with distribution function G and also independent of theX_i.One only observesLet S=1-F be survival function and S be the Kaplan-Meier estimator,i.e.,where Z_are the order statistics of Z_i and δ_((i))are the corresponping censoring indicatorfunctions.Define the density estimator of X_i by where =1-and h_n(>0)↓0.

STRONG UNIFORM CONSISTENCY FOR DENSITY ESTIMATOR FROM RANDOMLY CENSORED DATA

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))\]$.