Probability theory/Statistics
A note on the strong consistency of a constrained maximum likelihood estimator used in crash data modeling
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1147-1152.

In this note, we consider the Maximum Likelihood Estimator (MLE) of the vector parameter Φ=(θ,ϕT)T of dimension R (R>1) used in crash-data modeling where θ>0 and ϕ belongs to the simplex of order R1. We prove the strong consistency of this constrained estimator making capital out of the cyclic form between the components of the MLE.

Dans cette note, nous considérons l'estimateur du maximum de vraisemblance (EMV) du vecteur paramètre Φ=(θ,ϕT)T de dimension R (R>1) utilisé dans la modélisation des données d'accidents où θ>0 et ϕ appartient au simplexe d'ordre R1. Nous démontrons la consistance forte de cet estimateur sous contraintes en exploitant la forme cyclique entre les composantes de cet estimateur.

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DOI: 10.1016/j.crma.2015.09.025
Geraldo, Issa Cherif 1, 2; N'Guessan, Assi 2; Gneyou, Kossi Essona 1, 3

1 Département de mathématiques et informatique, Université catholique de l'Afrique de l'Ouest, Unité universitaire du Togo (UCAO–UUT), 01 B.P. 1502 Lomé 01, Lomé, Togo
2 Laboratoire Paul-Painlevé, UMR CNRS 8524, Université de Lille-1, 59655 Villeneuve d'Ascq cedex, France
3 Département de mathématiques, Faculté des sciences, Université de Lomé, B.P. 1515 Lomé, Togo
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Geraldo, Issa Cherif; N'Guessan, Assi; Gneyou, Kossi Essona. A note on the strong consistency of a constrained maximum likelihood estimator used in crash data modeling. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1147-1152. doi : 10.1016/j.crma.2015.09.025. http://www.numdam.org/articles/10.1016/j.crma.2015.09.025/

[1] Barvínek, E.; Daler, I.; Francu, J. Convergence of sequences of inverse functions, Arch. Math., Volume 27 (1991) no. 3–4, pp. 201-204

[2] Chafai, D.; Concordet, D. On the strong consistency of asymptotic M-estimators, J. Stat. Plan. Inference, Volume 137 (2007), pp. 2774-2783

[3] Chung, K.L. A Course in Probability Theory, Academic Press, 2001

[4] Ferguson, T.S. An inconsistent maximum likelihood estimate, J. Am. Stat. Assoc., Volume 77 (1982) no. 380, pp. 831-834

[5] Fiorin, S. The strong consistency for maximum likelihood estimates: a proof not based on the likelihood ratio, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2000), pp. 721-726

[6] Kourouklis, S. On the strong consistency of a solution to the likelihood equation, Stat. Probab. Lett., Volume 5 (1987), pp. 23-27

[7] Monahan, J.F. Numerical Methods of Statistics, Cambridge University Press, 2011

[8] N'Guessan, A. Analytical existence of solutions to a system of non-linear equations with application, J. Comput. Appl. Math., Volume 234 (2010), pp. 297-304

[9] N'Guessan, A.; Essai, A.; Langrand, C. Estimation multidimensionnelle des contrôles et de l'effet moyen d'une mesure de sécurité routière, Rev. Stat. Appl., Volume 49 (2001) no. 2, pp. 85-102

[10] N'Guessan, A.; Geraldo, I.C. A cyclic algorithm for maximum likelihood estimation using Schur complement, Numer. Linear Algebra Appl. (2015) | DOI

[11] N'Guessan, A.; Truffier, M. Impact d'un aménagement de sécurité routière sur la gravité des accidents de la route, J. Soc. Fr. Stat., Volume 149 (2008) no. 3, pp. 23-41

[12] Seo, B.; Lindsay, B.G. Nearly universal consistency of maximum likelihood in discrete models, Stat. Probab. Lett., Volume 83 (2013), pp. 1699-1702

[13] Strichartz, R.S. The Way of Analysis, Jones and Bartlett Books in Mathematics, Jones and Bartlett Publishers, 2000

[14] Van der Vaart, A.W. Asymptotic Statistics, Cambridge University Press, 1998

[15] Wald, A. Note on the consistency of the maximum likelihood estimate, Ann. Math. Stat., Volume 20 (1949) no. 4, pp. 595-601

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