We develop a test for spherical symmetry of a multivariate distribution $P$ that works even when the dimension of the data $d$ is larger than the sample size $n$. We propose a non-negative measure $\zeta(P)$ such that $\zeta(P)=0$ if and only if $P$ is spherically symmetric. We construct a consistent estimator of $\zeta(P)$ using the data augmentation method and investigate its large sample properties. The proposed test based on this estimator is calibrated using a novel resampling algorithm. Our test controls the Type-I error, and it is consistent against general alternatives. We also study its behaviour for a sequence of alternatives $(1-\delta_n) F+\delta_n G$, where $\zeta(G)=0$ but $\zeta(F)>0$, and $\delta_n \in [0,1]$. When $\lim\sup\delta_n<1$, for any $G$, the power of our test converges to unity as $n$ increases. However, if $\lim\sup\delta_n=1$, the asymptotic power of our test depends on $\lim n(1-\delta_n)^2$. We establish this by proving the minimax rate optimality of our test over a suitable class of alternatives and showing that it is Pitman efficient when $\lim n(1-\delta_n)^2>0$. Moreover, our test is provably consistent for high-dimensional data even when $d$ is larger than $n$. Our numerical results amply demonstrate the superiority of the proposed test over some state-of-the-art methods.

翻译：本文提出了一种针对多元分布$P$的球面对称性检验方法，该方法在数据维度$d$大于样本量$n$时仍然有效。我们构造了一个非负测度$\zeta(P)$，使得$\zeta(P)=0$当且仅当$P$具有球面对称性。通过数据增强方法，我们构建了$\zeta(P)$的一致性估计量并研究了其大样本性质。基于该估计量的检验采用新颖的重抽样算法进行校准。所提检验能够控制第一类错误，并对一般备择假设具有一致性。我们还研究了检验在序列备择假设$(1-\delta_n) F+\delta_n G$下的表现，其中$\zeta(G)=0$但$\zeta(F)>0$，且$\delta_n \in [0,1]$。当$\lim\sup\delta_n<1$时，对任意$G$，所提检验的检验功效随$n$增大趋近于1。然而若$\lim\sup\delta_n=1$，检验的渐近功效取决于$\lim n(1-\delta_n)^2$。我们通过证明所提检验在合适的备择假设类上具有极小极大最优速率，并证明当$\lim n(1-\delta_n)^2>0$时其具有Pitman有效性，建立了上述结论。此外，即使当$d$大于$n$时，所提检验对高维数据仍具有一致性。数值结果充分表明该检验优于现有先进方法。