  This paper considers the problem of estimating probability density functions
on the rotation group $SO(3)$. Two distinct approaches are proposed, one based
on characteristic functions and the other on wavelets using the heat kernel.
Expressions are derived for their Mean Integrated Squared Errors. The
performance of the estimators is studied numerically and compared with the
performance of an existing technique using the De La Vall\'ee Poussin kernel
estimator. The heat-kernel wavelet approach appears to offer the best
convergence, with faster convergence to the optimal bound and guaranteed
positivity of the estimated probability density function.
