Existence and strong consistency of maximum likelihood estimates for 1-dimensional exponential families (Q1916223)

The maximum likelihood method is perhaps the most common method of point estimation. Unfortunately, maximum likelihood estimators do not always exist and may not be consistent even when they exist. This paper concentrates on the broad class of 1-dimensional exponential families which include many of the most commonly used distributions, such as the Poissons, binomials, exponentials, etc. with various parameter sets. For such families, the maximum likelihood estimates of the parameter will be shown to exist in most cases. Moreover, the maximum likelihood estimates are stronlgy consistent whenever they exist.