论文标题
椭圆形分布的cramér-wold定理
A Cramér-Wold theorem for elliptical distributions
论文作者
论文摘要
根据Cramér和Wold的著名定理，如果$ P $和$ Q $是$ \ Mathbb {r}^d $的两个Borel概率措施，其Projections $ P_L，Q_L $ in Line $ l $ in $ \ \ \ \ m mathbb {r}^r}^d $满足$ p_l $ p_l = q_l = q_l $ p_l = q_l $ p_l $ p_l = q_l $，然后我们的主要结果是，如果$ p $和$ q $都是椭圆形分布，那么为了证明$ p = q $，仅需检查$ p_l = q_l $对于某个$（d^2+d）/2 $ lines $ l $的$ p_l = q_l $。此外，$（d^2+d）/2 $是最佳的。椭圆形分布类别包含高斯分布以及许多其他兴趣的多元分布。我们的定理与cramér-wold定理的其他变体形成鲜明对比，因为没有任何假设对$ p $和$ q $的矩有限。我们使用结果来得出椭圆形分布平等的统计检验，并对测试进行了小型模拟研究，并将其与文献的其他测试进行了比较。我们还提出了学习的应用（二进制分类），再次用一个小的模拟说明
According to a well-known theorem of Cramér and Wold, if $P$ and $Q$ are two Borel probability measures on $\mathbb{R}^d$ whose projections $P_L,Q_L$ onto each line $L$ in $\mathbb{R}^d$ satisfy $P_L=Q_L$, then $P=Q$. Our main result is that, if $P$ and $Q$ are both elliptical distributions, then, to show that $P=Q$, it suffices merely to check that $P_L=Q_L$ for a certain set of $(d^2+d)/2$ lines $L$. Moreover $(d^2+d)/2$ is optimal. The class of elliptical distributions contains the Gaussian distributions as well as many other multivariate distributions of interest. Our theorem contrasts with other variants of the Cramér-Wold theorem, in that no assumption is made about the finiteness of moments of $P$ and $Q$. We use our results to derive a statistical test for equality of elliptical distributions, and carry out a small simulation study of the test, comparing it with other tests from the literature. We also give an application to learning (binary classification), again illustrated with a small simulation
