Testing uniformity on high-dimensional spheres: The non-null behaviour of the Bingham test

IF 1.5 Q2 PHYSICS, MATHEMATICAL
C. Cutting, D. Paindaveine, Thomas Verdebout
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引用次数: 3

Abstract

Testing uniformity on the unit sphere of R is a fundamental problem in directional statistics. In the framework of axial data, the most classical test of uniformity is the Bingham [8] test. Remarkably, this test does not need any modification to meet asymptotically the target null size in high-dimensional scenarios where p = pn diverges to infinity with the sample size n. However, while the non-null asymptotic behaviour of the Bingham test is well understood in standard asymptotic scenarios where n diverges to infinity with p fixed, nothing is known on the power of this test in high dimensions, not even under standard parametric alternatives such as Watson distributions. In this work, we therefore study the non-null behaviour of the Bingham test in high dimensions. First, we consider a semiparametric class of alternatives that includes Watson alternatives and we derive a local asymptotic normality (LAN) property. An application of Le Cam’s third lemma reveals that the Bingham test is blind to the corresponding contiguous alternatives, though. By using martingale central limit theorems, we therefore study the non-null behaviour of the Bingham test under more severe alternatives. Far from restricting to the aforementioned semiparametric alternatives, our results cover a broad class of rotationally symmetric alternatives, which allows us to consider non-axial alternatives, too. In every distributional framework we consider, the “detection threshold” of the Bingham test is identified and a comparison with the classical test of uniformity for non-axial data, namely the Rayleigh [40] test, is made possible. In the framework of axial data, we derive a lower bound on the minimax separation rate and establish that the Bingham test is minimax rate-optimal in the class of Watson distributions. MSC 2010 subject classifications: Primary 62H11, 62F05; secondary 62E20.
高维球体均匀性测试:Bingham试验的非零行为
在R的单位球上检验均匀性是方向统计中的一个基本问题。在轴向数据的框架下,最经典的均匀性测试是Bingham[8]测试。值得注意的是,这个测试不需要任何修改来满足渐近目标零大小在高维情况下,p = pn发散到正无穷与样本大小n。然而,而非空宾厄姆的渐近行为测试很好理解标准的渐近场景n发散与p∞固定的,没有什么是在这个测试在高维度的力量,甚至在标准沃森分布等参数的选择。因此,在这项工作中,我们研究了高维Bingham检验的非零行为。首先,我们考虑了一类包含Watson选项的半参数选项,并导出了一个局部渐近正态性(LAN)性质。勒卡姆第三引理的一个应用表明,宾厄姆检验对相应的相邻选择是盲目的。因此,我们利用鞅中心极限定理,研究了Bingham检验在更严格的替代条件下的非零性。我们的结果不仅局限于上述的半参数替代方案,还涵盖了广泛的旋转对称替代方案,这也允许我们考虑非轴向替代方案。在我们考虑的每一个分布框架中,我们确定了Bingham检验的“检测阈值”,并将其与非轴向数据均匀性的经典检验即Rayleigh[40]检验进行了比较。在轴向数据的框架下,我们导出了极小极大分离率的下界,并证明了Bingham检验在Watson分布中是极小极大分离率最优的。MSC 2010学科分类:初级62H11, 62F05;二次62 e20。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.30
自引率
0.00%
发文量
16
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