Journal of Global Optimization 21: 223–237, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands. 223 Finding GM-estimators with global optimization techniques RAFAEL BLANQUERO, EMILIO CARRIZOSA and EDUARDO CONDE Departamento de Estadística e Investigación Operativa, Universidad de Sevilla, Sevilla, Spain (e-mail: rblanque@cica.es; ecarriz@cica.es; educon@cica.es) Abstract. In this note we address the problem of finding the GM-estimator for the location parameter of a univariate random variable. When this problem is non-convex but d.c. one can use a standard covering method, which, in the one-dimensional case has a simple form. In this paper we exploit the structure of the problem in order to obtain d.c. decompositions with certain optimality proper- ties in the application of the algorithm. Numerical results show that this general-purpose algorithm outperforms previous ad-hoc methods for this problem. Key words: GM-estimators, Robust estimation, D.C. optimization, Covering methods 1. The model Given a sample of n observations y 1 ,y 2 ,...,y n the determination of a parameter that, in some sense, represents the data is a classical problem in Statistics. In the last decades the traditional least-squares method has been more and more frequently replaced by other approaches with better properties of robustness [19]. In particular, an M-estimator [14, 15], is an optimal solution of an optimization program of the form inf θ ∈R  1j n ρ(r j (θ)), (1) where (r 1 (θ),...,r n (θ)) is the vector of residuals, r j (θ) = y j - θ j = 1, 2,...,n and ρ : R -→ R is some continuous, even and nondecreasing function in R + . The class of M-estimators has been further enlarged to the class of so-called GM-estimators (generalized M-estimators), in which the influence of each residual is made dependent on the observation y j . In other words, θ ∗ is said to be a GM- estimator if it solves an optimization problem of the form inf θ ∈R σ(θ), (2) with σ(θ) =  1j n ρ j (θ),