Goodness-of-fit test for noisy directional data

We consider spherical data $X_i$ noised by a random rotation $\varepsilon_i\in$ SO(3) so that only the sample $Z_i=\varepsilon_iX_i$, $i=1,\dots, N$ is observed. We define a nonparametric test procedure to distinguish $H_0:$ ''the density $f$ of $X_i$ is the uniform density $f_0$ on the sphere'' and $H_1:$ ''$\|f-f_0\|_2^2\geq \C\psi_N$ and $f$ is in a Sobolev space with smoothness $s$''. For a noise density $f_\varepsilon$ with smoothness index $\nu$, we show that an adaptive procedure (i.e. $s$ is not assumed to be known) cannot have a faster rate of separation than $\psi_N^{ad}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2\nu+1)}$ and we provide a procedure which reaches this rate. We also deal with the case of super smooth noise. We illustrate the theory by implementing our test procedure for various kinds of noise on SO(3) and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.