Statistics, Vol. 39, No. 4, August 2005, 303–314 A covariance components estimation procedure when modelling a road safety measure in terms of linear constraints ASSI N’GUESSAN*† and CLAUDE LANGRAND‡ †Ecole Polytechnique Universitaire de Lille et, Laboratoire Paul Painlevé CNRS, UMR 8542, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France ‡Laboratoire Paul Painlevé CNRS, UMR 8542, U.F.R de Mathématiques pures et appliquées, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France (Received 15 December 2003; revised 7 December 2004; in final form 9 March 2005) This paper deals with the asymptotic estimation of the covariance between the average effect of a road safety measure and the accident risks when the multidimensional parameter of interest is constrained. The explicit asymptotic covariance is achieved through Schur complements technical method. This approach also provides a powerful tool to formally compute the explicit covariance between two different accident risks whatever the experimental sites. The easiness of the obtention of the asymptotic covariances without matrix inversion is an attractive aspect of this new approach. Some examples of formal estimation are discussed to back up the method. Keywords: Road safety measure; Accident data; Multinomial model; Constrained maximum like- lihood; Fisher information matrix; Schur complements; Asymptotic covariance matrix; Formal estimation AMS Classification: 62F10; 62F12; 62F25; 62F30 1. Introduction The purpose of this paper is to give a formal and analytical expression of the asymptotic covariance between the components of interest of a constrained vector parameter arising in a linear problem of restricted maximum likelihood estimation (RMLE) of road safety measure modelling. The general framework for RMLE is well-known and is not discussed here. Aitchison and Silvey [1] consider the case of i.i.d. observations along the lines due to Cramér [2] and prove, under certain conditions, asymptotic existence, consistency and normality of the RMLE. Silvey [3] considers the i.i.d. case, but allowing the Fisher information matrix to be singular. He uses the alternative approach to prove consistency due toWald [4]. Magnus [5] considers an alternative way of dealing with RMLE asymptotic results in the case of non i.i.d. observations using linear structures properties. Crowder [6] considers the case of non i.i.d. *Corresponding author. Email: assi.nguessan@polytech-lille.fr; Fax: +33 3 28 76 73 01. Statistics ISSN 0233-1888 print/ISSN 1029-4910 online © 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/02331880500108544