On the power of axial tests of uniformity on spheres. (English) Zbl 1442.62113

In the setting of directional data, that is, data distributed on the unit hypersphere \[ \mathcal{S}^{p-1}=\{\mathbf{x}\in\mathbb{R}^p : \|\mathbf{x}\|^2=1\} \] in \(\mathbb{R}^p\), the authors consider the problem of testing the null hypothesis of uniformly distributed data in cases where the data are axial, i.e., the data come from a distribution that has equal density on antipodal points. Alternative hypotheses are related to a semiparametric family of densities specified through a function \(f\), a vector \(\boldsymbol{\theta}\in\mathcal{S}^{p-1}\), and a concentration parameter \(\kappa\) (where \(\kappa=0\) gives a uniform distribution, regardless of the choice of \(\boldsymbol{\theta}\) or \(f\)).
In this setting, the authors first consider the case where \(\boldsymbol{\theta}\) is known, for which they prove a local asymptotic normality result which shows that a given test based on the sample covariance matrix of the data is locally asymptotically most powerful. Then, in the case where \(\boldsymbol{\theta}\) is unknown, the authors investigate properties of test statistics including that proposed by C. Bingham [Ann. Stat. 2, 1201–1225 (1974; Zbl 0297.62010)] and ones that take advantage of the ‘single-spiked’ nature of the alternatives under consideration. Simulations are used throughout to verify and illustrate the results obtained.


62H11 Directional data; spatial statistics
62E17 Approximations to statistical distributions (nonasymptotic)
62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
62R30 Statistics on manifolds


Zbl 0297.62010


Full Text: DOI arXiv Euclid


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