Goodness-of-fit test for noisy directional data

Abstract

We consider spherical data $X_{i}$ noised by a random rotation $\varepsilon_{i}\in\operatorname{SO} (3)$ so that only the sample $Z_{i}=\varepsilon_{i}X_{i}$, $i=1,\dots,N$ 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 $H_{1}$: “$\|f-f_{0}\|_{2}^{2}\geq\mathcal{C} \psi_{N}$ 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 $\psi_{N}^{\mathrm{ad}}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2\nu+1)}$ 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 $\operatorname{SO}(3)$ and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.