Summary: We obtain approximations to the distribution of the exponent in the matrix Fisher distributions on and on whose density with respect to Haar measure is proportional to . Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to , on the unit sphere in Euclidean - dimensional space. The matrix Fisher distribution arises as the exact conditional distribution of the maximum likelihood estimate of the unknown orthogonal matrix in the spherical regression model on with Fisher distributed errors. It also arises as the exact conditional distribution of the maximum likelihood estimate of the unknown orthogonal matrix in a model of Procrustes analysis in which location and orientation, but not scale, changes are allowed.
These methods allow determination of a confidence region for the unknown rotation for moderate sample sizes with moderate error concentrations when the error concentration parameter is known.