On the consistency of the maximum likelihood estimator for the three parameter lognormal distribution

The three parameter log-normal distribution is a popular non-regular model, but surprisingly, whether the local maximum likelihood estimator (MLE) for parameter estimation is consistent or not has been speculated about since the 1960s. This note gives a rigorous proof for the existence of a consistent MLE for the three parameter log-normal distribution, which solves a problem that has been recognized and unsolved for 50 years. Our results also imply a uniform local asymptotic normality condition for the three parameter log-normal distribution. In addition, we give results on the asymptotic normality and the uniqueness of the local MLE.