  This paper extends our earlier article, "Computing the confidence levels for
a root-mean-square test of goodness-of-fit;" unlike in the earlier article, the
models in the present paper involve parameter estimation -- both the null and
alternative hypotheses in the associated tests are composite. We provide
efficient black-box algorithms for calculating the asymptotic confidence levels
of a variant on the classic chi-squared test. In some circumstances, it is also
feasible to compute the exact confidence levels via Monte Carlo simulation.
