## 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.

### MSC:

 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

rotasym
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### References:

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