A new hybrid algorithm for maximum likelihood estimation in a model of accident frequencies

Abstract

In this paper, we are interested in the numerical computation of the constrained maximum likelihood estimator (MLE) of the parameter vector of a discrete statistical model used in statistics applied to road safety. The parameter vector is divided into two blocks: one block with the parameter of interest and the second block with secondary parameters. The MLE is the solution to a system of non-linear implicit equations difficult to solve in closed-form. To overcome this difficulty, we propose a hybrid algorithm (HA) mixing the use of a one-dimensional Newton-Raphson (NR) algorithm for the first equation of the system and a fixed-point strategy for the remaining equations. Our proposed algorithm involves no matrix inversion but it partially enjoys the quadratic convergence rate of the one-dimensional NR algorithm. We illustrate its performance on simulated data and we compare it to Newton-Raphson (NR) and quasi-Newton algorithms which are two of the most used optimization algorithms. The results suggest that our HA outperforms NR and quasi-Newton algorithms. It is accurate and converges quickly for all the starting values.