Abstract: Foutz (1977) uses the Inverse Function Theorem to prove the existence of a unique and consistent solution to the likelihood equations. This note extends his results in three useful directions. The first is to remark that with minor modification the same proof may be used to show that the solution to the likelihood equations converges asymptotically to the least-false parameter (Hjort (1986, 1992)) when the true probability distribution of the data differs from the parametric family of models under consideration. The second is to extend his results to certain situations in which the dimension of the parameter space is not fixed but expanding at some rate less than the sample size. Lastly, we indicate how this result may be applied in more general ,M-estimation problems. An application of these results to proving consistency in problems involving splines is discussed.
Key words and phrases: Least-false parameter, maximum likelihood, regression spline, uniform convergence.