Suppose $u_1, \cdots, u_n, v_1, \cdots, v_n$ are random points on the sphere such that for unknown points $\xi_1, \cdots, \xi_n$ and unknown rotation $A_0$, the distribution of $u_i$ depends only on $u^t_i\xi_i$ and that of $v_i$ on $v^t_iA_0\xi_i$. This paper provides asymptotic tests and confidence regions for $A_0$ and for its axis of rotation. Results are given in arbitrary dimension.
"Spherical Regression with Errors in Variables." Ann. Statist. 17 (1) 293 - 306, March, 1989. https://doi.org/10.1214/aos/1176347017