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This paper introduces two new families of non-parametric tests ofgoodness-of-fit on the compact classical groups. One of them is a family oftests for the eigenvalue distribution induced by the uniform distribution,which is consistent against all fixed alternatives. The other is a family oftests for the uniform distribution on the entire group, which is againconsistent against all fixed alternatives. We find the asymptotic distributionunder the null and general alternatives. The tests are proved to beasymptotically admissible. Local power is derived and the global properties ofthe power function against local alternatives are explored.  The new tests are validated on two random walks for which the mixing-time isstudied in the literature. The new tests, and several others, are applied tothe Markov chain sampler proposed by \cite{jones2011randomized}, providingstrong evidence supporting the claim that the sampler mixes quickly.

New Tests of Uniformity on the Compact Classical Groups as Diagnostics for Weak-star Mixing of Markov Chains