Exact upper and lower bounds on the ratio $\mathsf{E}w(\mathbf{X}-\mathbf{v})/\mathsf{E}w(\mathbf{X})$ for a centered Gaussian random vector $\mathbf{X}$ in $\mathbb{R}^n$, as well as bounds on the rate of change of $\mathsf{E}w(\mathbf{X}-t\mathbf{v})$ in $t$, where $w\colon\mathbb{R}^n\to[0,\infty)$ is any even unimodal function and $\mathbf{v}$ is any vector in $\mathbb{R}^n$. As a corollary of such results, exact upper and lower bounds on the power function of statistical tests for the mean of a multivariate normal distribution are given.
翻译:以$\mathb{R ⁇ }{X}w(\mathbf{X}-\mathbbf{v}}})/\mathsf{E}w(\mathbf{X})为核心的高斯随机矢量 $\mathb{R}{X}美元 的精度和下限,以及以$\mathsf{E}}w(\mathbf{X}X}-t\mathb{f{v}}为单位的精确值,其中$w\cron\mathb{R}}n\\\\\\\\\\\[0,\infty}美元甚至是一个单式函数,$\mathbf{v}美元是以$mathb{R{n$为单位的任何矢量。作为结果的必然结果,在多变量正常分布平均值的统计测试功率上给出了精确的上下限和下限。