Existence, uniqueness and strong consistency of the maximum likelihood estimator for a model of accidents frequencies

Abstract

The aim of this paper is to prove the existence, uniqueness and strong consistency (i.e. almost sure convergence to the true unknown value) of the maximum likelihood estimator (MLE) of the vector parameter for a statistical model used in statistics applied to road safety. In the general case, the strong consistency of the MLE may be established by using the well-known result by Abraham Wald (in 1949) or its variants under a set of conditions. However, for the model considered in this paper, all these conditions are very difficult to verify because of the great dimension of the parameter space and the rather complex expression of the log-likelihood function. To circumvent these difficulties, we first demonstrate that the MLE exists and is unique afterwards we demonstrate the strong consistency of the MLE using the properties of the model and some theorems of mathematical analysis.