Goodness-of-fit test for noisy directional data

We consider spherical data Xi noised by a random rotation εi ∈ SO(3) so that only the sample Zi = εiXi, i = 1, . . . , N is observed. We define a nonparametric test procedure to distinguish H0 : ”the density f of Xi is the uniform density f0 on the sphere” and H1 : ”‖f − f0‖2 ≥ CψN and f is in a Sobolev space with smoothness s”. For a noise density fε with smoothness index ν, we show that an adaptive procedure (i.e. s is not assumed to be known) cannot have a faster rate of separation than ψ N (s) = (N/ √ log log(N))−2s/(2s+2ν+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.