The MLE of Aigner, Amemiya, and Poirier is not the Expectile MLE

This article compares two asymmetric Gaussian likelihood models and their corresponding estimators. Recently, there has been confusion in the literature regarding these models and: (1) whether they are the same, or (2) whether both of them can be used to estimate expectiles. After the comparison, it becomes clear that they are not the same and only one of these models is appropriate for that purpose. The similarity between these models is purely superficial. The historical origin of expectiles has also been disputed: some degree of credit can be shared between two papers