On a criticism of the profile likelihood function

Abstract

The profile likelihood function is often criticized for giving strange or unintuitive results. In the cases discussed here these are due to the use of density functions that have singularities. These singularities are naturally inherited by the profile likelihood function. It is therefore apparently important to be reminded that likelihood functions are proportional to probability functions, and so cannot have singularities. When this issue is addressed, then the profile likelihood poses no problems of this sort. This is of particular importance since the profile likelihood is a commonly used method for dealing with separate estimation of parameters.