论文标题
关于基于球差异的一些两样本测试的高维行为
On High Dimensional Behaviour of Some Two-Sample Tests Based on Ball Divergence
论文作者
论文摘要
在本文中，我们提出了一些基于球差异的两样本测试，并研究了它们的高维行为。首先，我们研究了它们的行为，用于高维，低样本量（HDLSS）数据，在适当的规律性条件下，我们在HDLSS制度中建立了它们的一致性，在HDLSS制度中，数据的尺寸增长到无穷大，而来自两个分布的样本大小则保持固定。此外，我们表明，当样本量也随尺寸增加时，可以放松这些条件，在这种情况下，即使缩小替代方案也可以证明一致性。我们使用一个涉及两个正常分布的简单示例来证明，即使HDLSS制度中没有一致的测试，如果样本量以适当的速度随维度增加而增加，则提议的测试的功能也会融合到Unity。通过在一系列替代方案上建立我们的测试的最小值最佳来获得此速率。分析了几个模拟和基准数据集，以将这些提出的测试的性能与可用于测试两个高维概率分布的相等性的最新方法进行比较。
In this article, we propose some two-sample tests based on ball divergence and investigate their high dimensional behavior. First, we study their behavior for High Dimension, Low Sample Size (HDLSS) data, and under appropriate regularity conditions, we establish their consistency in the HDLSS regime, where the dimension of the data grows to infinity while the sample sizes from the two distributions remain fixed. Further, we show that these conditions can be relaxed when the sample sizes also increase with the dimension, and in such cases, consistency can be proved even for shrinking alternatives. We use a simple example involving two normal distributions to prove that even when there are no consistent tests in the HDLSS regime, the powers of the proposed tests can converge to unity if the sample sizes increase with the dimension at an appropriate rate. This rate is obtained by establishing the minimax rate optimality of our tests over a certain class of alternatives. Several simulated and benchmark data sets are analyzed to compare the performance of these proposed tests with the state-of-the-art methods that can be used for testing the equality of two high-dimensional probability distributions.
