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.

Exact computation of the asymptotic variance matrix for a tuple of multinomial distributions with dependent parameters

“…, s, ( In the general case s > 1, the resolution of (5.1) under the constraints (5.2) -(5.3) can only be done numerically. In this context, it has been proven that this problem is difficult to deal with for classical optimization algorithms when the number of parameters increases and therefore, two optimization algorithms have been developed and their efficiency has be proven on simulated data [6,8]. The results obtained in our present paper therefore complete those in [6,8] and allow to make applications on real data in the case s > 1 as the authors in [7] did for s = 1.…”

A Fast and Efficient Estimation of the Parameters of a Model of Accident Frequencies via an MM Algorithm

“…We compare, in R software [18], our MM algorithm and its accelerated version SqS3 to the NR algorithm (package nleqslv [19]) and quasi-Newton BFGS algorithm (package alabama [20]). The design of the simulation study is inspired from [21,22].…”