Summary: Applications are given of a formula for the exact probability density function of the maximum likelihood estimates of a statistical model, where the data generating model is allowed to differ from the estimation model. The main examples are supported by simulation experiments. Curved exponential families are investigated, for which an approach is described that can be used in many practical situations. The distribution of a maximum likelihood estimator in exponential regression is developed. Nonlinear regression is then considered, with an example of a model discrepancy situation arising in ELISA immunoassays and similar biochemical titrations. An incorrect logistic model is specified for a titration curve that is used for describing the reaction of a chemical sample to applied substrate concentration. A method is suggested to reduce the amount of bias in the estimate of binding affinity. Finally, there is a prospective discussion of other possible uses of the technique, including general comparisons of sets of alternative models in frequentist and Bayesian settings, applications to robust estimation and extensions beyond maximum likelihood estimates.