基于方向数据的核密度估计的对称检验的大偏差

Large Deviations for a Test of Symmetry Based on Kernel Density Estimator of Directional Data

中文关键词:  对称检验  核密度估计  方向数据  大偏差

英文关键词:symmetry test  kernel density estimator  directional data  large deviations

中文摘要:

      设$f_n$是基于核函数$K$和取值于$d$-维单位球面${\mathbb{S}}^{d-1}$的独立同分布随机变量列的非参数核密度估计. 我们证明了若核函数是有界变差函数, 随机变量的密度函数$f$是连续的和对称的, $\{\sup_{x\in {\mathbb{SS}}^{d-1}}|f_n(x)-f_n(-x)|,n\ge 1\}$的大偏差原理成立.

英文摘要:

      Assume that $f_n$ is the nonparametric kernel density estimator of directional data based on a kernel function $K$ and a sequence of independent and identically distributed random variables taking values in $d$-dimensional unit sphere ${\mathbb{S}}^{d-1}$. We established that the large deviation principle for $\{\sup_{x\in {\mathbb{S}}^{d-1}}|f_n(x)-f_n(-x)|,n\ge 1\}$ holds if the kernel function is a function with bounded variation, and the density function $f$ of the random variables is continuous and symmetric.