Summary: Two distinct problems can arise in maximum likelihood estimation in certain three-parameter problems. One problem, associated with the fitting of a threshold parameter, occurs if the fitted distribution has to be very positively skewed. This problem is well known. However, there is a second and unrelated difficulty which occurs quite often in practice and which arises when the distribution is not at all skew. This problem is not so well understood.
It is shown in this paper that, within the three-parameter model, there is an embedded two-parameter special case which corresponds to infinite parameter values in the original model. The problem arises when this embedded model is the best fit to the data. The problem is shown to be easily resolved by first carrying out a check to see whether the embedded model should be fitted instead of the three-parameter model. Formal tests for this are discussed. The gamma, inverse Gaussian, log-normal and Weibull distributions are examples where the problem occurs. Numerical examples are provided for illustration.