Existence of maximum likelihood estimation for three-parameter log-normal distribution

In the literature concerning maximum likelihood estimation (MLE), likelihood is always defined as the product of corresponding density function for the observations. When the distribution function is log-normal, the unboundedness problem will lead to nonexistence of the estimation. In this article, we introduce an observed error term to modify the traditional likelihood function and consider the condition for the existence of the MLEs of three parameters in the corrected likelihood function. It is shown through an example that the MLEs of three parameters exists if the condition is satisfied and the estimation is not sensitive to the choice of the observation error.