It is shown that, under usual regularity conditions, the maximum likelihood estimator of a structural parameter is strongly consistent, when the (infinitely many) incidental parameters are independently distributed chance variables with a common unknown distribution function. The latter is also consistently estimated although it is not assumed to belong to a parametric class. Application is made to several problems, in particular to the problem of estimating a straight line with both variables subject to error, which thus after all has a maximum likelihood solution.
"Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters." Ann. Math. Statist. 27 (4) 887 - 906, December, 1956. https://doi.org/10.1214/aoms/1177728066