On the multivariate runs test
On the multivariate runs test
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DOI：
10.1214/aos/1018031112
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发表时间：
1999-03
影响因子：
4.5
通讯作者：
N. Henze;M. Penrose
中科院分区：
文献类型：
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作者：
N. Henze;M. Penrose
对于具有公共密度$f$和公共密度$g$的独立$d$-变量随机变量$X_1，\dots，X_m $和$Y_1，\dots，Y_n $，设$R_{m，n}$是连接不同样本点的顶点为$X_1，\dots，X_m $，$Y_1，\dots，Y_n $的最小生成树中的边数。Friedman和Rafsky指出，对于R_{m，n}$的小值，拒绝H_0 $的$H_0：f = g$的检验应该有能力反对一般的替代方案。本文证明了R_{m，n}$在H_0 $下是渐近分布自由的，并证明了基于R_{m，n}$的多元两样本检验是普适相合的。
For independent $d$-variate random variables $X_1,\dots,X_m$ with common density $f$ and $Y_1,\dots,Y_n$ with common density $g$, let $R_{m,n}$ be the number of edges in the minimal spanning tree with vertices $X_1,\dots,X_m$, $Y_1,\dots,Y_n$ that connect points from different samples. Friedman and Rafsky conjectured that a test of $H_0: f = g$ that rejects $H_0$ for small values of $R_{m,n}$ should have power against general alternatives. We prove that $R_{m,n}$ is asymptotically distribution-free under $H_0$ , and that the multivariate two-sample test based on $R_{m,n}$ is universally consistent.