A Distribution-Free Multivariate Sign Test Based on Interdirections
提出一种称为方向间距离的计数指标，用于测量观测向量间的角距离，并基于此构建无分布多元符号检验，适用于椭圆对称和偏态分布，在小样本下具有无分布性质，且渐近服从卡方分布。
Abstract Distribution-free tests are investigated for the one-sample multivariate location problem. Counts, called interdirections, which measure the angular distance between two observation vectors relative to the positions of the other observations, are introduced. These counts are invariant under nonsingular linear transformations and have a small-sample distribution-free property over a broad class of population models, called distributions with elliptical directions, which includes all elliptically symmetric populations and many skewed populations. A sign test based on interdirections is described, including, as special cases, the two-sided univariate sign test and Blumen's bivariate sign test. The statistic is shown to have a limiting χ2 p null distribution and, because it is based on interdirections, it is also seen to be invariant and to have a small-sample distribution-free property. Pitman asymptotic relative efficiencies and a Monte Carlo study show the test to perform well compared with Hotelling's T 2, particularly when the underlying population is heavy-tailed or skewed. In addition, it consistently outperforms the component sign test, which is often recommended in the nonparametric literature. Key Words: Location testOne-sample multivariate location