On the existence and uniqueness of the maximum likelihood estimates for certain generalized linear models

Generalized linear models were defined by Nelder & Wedderburn (1972) and include a large class of useful statistical models. It is shown that for some of these models maximum likelihood estimates always exist and that for some others they exist provided certain degeneracies in the data do not occur. Similar results are obtained for uniqueness of maximum likelihood estimates and for their being confined to the interior of the parameter space. For instance, with the familiar model of probit analysis it is shown that under certain conditions the maximum likelihood estimate exists, and that if it does exist it is unique. The models considered also include those involving the normal, Poisson and gamma distributions with power and logarithmic transformations to linearity.