While spherical distributions have been used in many statistical models for high-dimensional data analysis, there are few easily implemented statistics for testing spherical symmetry for the underlying distribution of high- dimensional data. Many existing statistics for this purpose were constructed by the theory of empirical processes and turn out to converge slowly to their lim- iting distributions. Some existing statistics for the same purpose were given in the form of high-dimensional integrals that are not easily evaluated in numer- ical computation. In this paper, we develop some necessary tests for spherical symmetry based on both univariate and multivariate uniform statistics. These statistics are easily evaluated numerically and have simple limiting distribu- tions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and power.
An omnibus test for spherical symmetry in ℝ2 is proposed, employing localized empirical likelihood. The thus obtained test statistic is distribution free under the null hypothesis. The asymptotic null distribution is established and critical values for typical sample sizes, as well as the asymptotic ones, are presented. In a simulation study, the good performance of the test is demonstrated. Furthermore, a real data example is presented.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.