The stationary distribution of allele frequencies under a variety of
Wright--Fisher $k$-allele models with selection and parent independent mutation
is well studied. However, the statistical properties of maximum likelihood
estimates of parameters under these models are not well understood. Under each
of these models there is a point in data space which carries the strongest
possible signal for selection, yet, at this point, the likelihood is unbounded.
This result remains valid even if all of the mutation parameters are assumed to
be known.