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In this paper, we consider a coherent theory about the asymptotic representations for a family of inequality indices called Theil-Like Inequality Measures (TLIM ), within a Gaussian field. The theory uses the functional empirical process approach. We provide the finite-distribution and uniform asymptotic normality of the element of the TLIM class in a unified approach rather than in a case by case one. The results are then applied to some UEMOA countries databases. A Theil-like class of inequality measures, its asymptotic normality theory and applications 1700 Résumé. (French) Dans cet article, nous présentons une théorie cohérente de représentations asymptotiques d'une famille de mesures d'inégalité dénommée TLIM dans un champ gaussien précis. Notre méthode est fondée sur le processus empirique fonctionnel. Nous tirons de la représentation asymptotique les limites en distribution des estimateurs plug-in des membres de la famille en dimension finie. Les résultats sont ensuite appliquésà des données issues des pays de l'UEMOA.

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.

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