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The canonical way to do the test is to perform the spherical harmonic transform of the empirical distribution and then check that the power spectrum decays, but this is presumably fairly expensive. Is there some more efficient way (I am currently most interested in $S^2$ in $\mathbb{R}^3,$ but certainly something that scales reasonably as the dimension goes up would be good (the spherical harmonic method does not))?

kjetil b halvorsen
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Igor Rivin
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  • Please see https://stats.stackexchange.com/a/7984/919 for one approach. But you could apply (literally) any equal-area mapping and test for uniformity of the image. The equal-area cylindrical map (send each point to its longitude and distance from the Equator) is attractive because it maps the sphere to a rectangle. – whuber Mar 04 '23 at 00:23
  • @whuber Well, yes, one can randomly rotate the sphere and then check if the x-coordinate is uniform. But there is an obvious technical question there: if the p-value of the hypothesis "a random projection is uniform" is $\alpha$ for $k$ random projections, what is the $p$-value of the hypothesis that the original distribution is uniform on $S^2?$ – Igor Rivin Mar 04 '23 at 00:50
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    Do you need to be able to detect all possible departures or is it ok to have power against just some important set? For example, using the sum of the 3-vectors to the points as a test statistic will have good power against distributions that concentrate near one point or axis, or that avoid one point or axis, but it won't do well against distributions that, say, cluster near vertices of an eicosahedron. – Thomas Lumley Mar 04 '23 at 08:31
  • I am not discussing random coordinate projections. The only special thing you need to do is map the sphere onto a patch using an equal-area mapping. The question then is equivalent to testing the image of your empirical distribution for uniformity. – whuber Mar 04 '23 at 13:24
  • On the sphere, entropy is maximized by the uniform distribution. So if you can estimate the (differential) entropy, a test could be built on that? Here (arXiv) is a paper looking at many methods and there is an R implementation: https://cran.r-project.org/web/packages/sphunif/vignettes/sphunif.html – kjetil b halvorsen Mar 04 '23 at 22:47

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