Abstract
Motivated by the central role played by rotationally symmetric distributions in directional statistics, we consider the problem of testing rotational symmetry on the hypersphere. We adopt a semiparametric approach and tackle problems where the location of the symmetry axis is either specified or unspecified. For each problem, we define two tests and study their asymptotic properties under very mild conditions. We introduce two new classes of directional distributions that extend the rotationally symmetric class and are of independent interest. We prove that each test is locally asymptotically maximin, in the Le Cam sense, for one kind of the alternatives given by the new classes of distributions, for both specified and unspecified symmetry axis. The tests, aimed to detect location- and scatter-like alternatives, are combined into convenient hybrid tests that are consistent against both alternatives. We perform Monte Carlo experiments that illustrate the finite-sample performances of the proposed tests and their agreement with the asymptotic results. Finally, the practical relevance of our tests is illustrated on a real data application from astronomy. The R package rotasym implements the proposed tests and allows practitioners to reproduce the data application. Supplementary materials for this article are available online.
Highlights
1.1 MotivationDirectional statistics deals with data belonging to the unit hypersphere Sp−1 := {x ∈ Rp : x 2 = xT x = 1} of Rp
The most popular parametric model in directional statistics, which can be traced back to the beginning of the 20th century, is the von Mises–Fisher model characterized by the density x → cM p,κ exp(κ xT θ), where θ ∈ Sp−1 is a location parameter, κ > 0 is a concentration parameter, and cM p,κ is a normalizing constant
We introduce two new classes C1 and C2 of distributions on Sp−1 that are of independent interest and that may serve as natural alternatives to rotational symmetry
Summary
For p ≥ 3, the problem is much more difficult, which explains that the corresponding literature is much sparser: to the best of our knowledge, for p ≥ 3, only Jupp and Spurr (1983) and Ley and Verdebout (2017) addressed the problem of testing rotational symmetry in a semiparametric way (i.e., without specifying the function g) The former considered a test for symmetry in dimension p ≥ 2 using the Sobolev tests machinery from Giné (1975), whereas the latter established the efficiency of the Watson (1983) test against a new type of non-rotationally symmetric alternatives. Figueiredo (2012) considered goodness-of-fit tests for vMF distributions, while Boente et al (2014) introduced goodness-of-fit tests based on kernel density estimation for any (possibly non-rotationally symmetric) distribution
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
