Circular unitary ensembles: parametric models and their asymptotic maximum likelihood estimates

Parametrized families of distributions for the circular unitary ensemble in random matrix theory are considered which are connected to Toeplitz determinants and which have many applications in mathematics (for example to the longest increasing subsequences of random permutations) and physics (for example to nuclear physics and quantum gravity). We develop a theory for the unknown parameter estimated by an asymptotic maximum likelihood estimator, which, in the limit, behaves as the maximum likelihood estimator if the latter is well defined and the family is sufficiently smooth. They are asymptotically unbiased and normally distributed, where the norming constants are unconventional because of long range dependence.