On Limit Distribution of Maximal Deviation of Empirical Distribution Density and Regression Function. II
On Limit Distribution of Maximal Deviation of Empirical Distribution Density and Regression Function. II
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经验分布密度最大偏差极限分布与回归函数。
DOI：
10.1137/s0040585x97985029
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发表时间：
2011
影响因子：
0.6
通讯作者：
M. Muminov
中科院分区：
文献类型：
--
作者：
M. Muminov
In this paper, for an unknown distribution function $f(t)$, $t\in {\bf R}^\nu$, a random vector $X\in {\bf R}^{\nu}$, and a regression function $r(t)={\bfE}\,(Y\,|\,X=t)$ of a random vector $(X,Y)$, $X\in {\bf R}^{\nu}$, $Y\in {\bf R}^{1}$, nonparametric kernel estimates $f_n(t)$ and $r_n(t)$ are constructed. It is proved that distribution of the maximal deviation of these estimators from the true distribution density $f(t)$ and the regression function $r(t)$ tend to the double exponential law as ${n \rightarrow \infty}$. With the aid of the constructed estimators we find a confidence region for $f(t)$ and $r(t)$, corresponding to the given confidence coefficient $\alpha$ $(0<\alpha <1)$, and construct a criterion for testing the hypothesis $H_0: f(t)=f_0(t)$ (respectively, $H_0': r(t)=r_0(t))$, where $f_0(t)$ is a given a priori distribution density, and $r_0(t)$ is a given function.
In this paper, for an unknown distribution function $f(t)$, $t\in {\bf R}^\nu$, a random vector $X\in {\bf R}^{\nu}$, and a regression function $r(t)={\bfE}\,(Y\,|\,X=t)$ of a random vector $(X,Y)$, $X\in {\bf R}^{\nu}$, $Y\in {\bf R}^{1}$, nonparametric kernel estimates $f_n(t)$ and $r_n(t)$ are constructed. It is proved that distribution of the maximal deviation of these estimators from the true distribution density $f(t)$ and the regression function $r(t)$ tend to the double exponential law as ${n \rightarrow \infty}$. With the aid of the constructed estimators we find a confidence region for $f(t)$ and $r(t)$, corresponding to the given confidence coefficient $\alpha$ $(0<\alpha <1)$, and construct a criterion for testing the hypothesis $H_0: f(t)=f_0(t)$ (respectively, $H_0': r(t)=r_0(t))$, where $f_0(t)$ is a given a priori distribution density, and $r_0(t)$ is a given function.