Journal of Statistical Planning and Inference 136 (2006) 1281 – 1301 www.elsevier.com/locate/jspi The contribution of the maximum to the sum of excesses for testing max-domains of attraction Cláudia Neves a , 1 , Jan Picek b, 2 , M.I. Fraga Alves c , ∗, 1 a UIMA, Department of Mathematics, University of Aveiro, Portugal b Department of Applied Mathematics, Technical University of Liberec, Czech Republic c CEAUL, DEIO, Faculty of Sciences, University of Lisbon, Portugal Received 1 July 2003; accepted 16 September 2004 Available online 26 November 2004 Abstract We consider an i.i.d. sample, from an underlying distribution function with unknown shape, location and scale parameters, belonging to some max-domain of attraction. We study the performance of a test statistic which is merely a ratio between the maximum and the mean of the sample of the excesses above some random threshold. This scale/location invariant ratio turns out to be very useful in the construction of an asymptotically size  test for the null hypothesis that the distribution comes from the Gumbel domain of attraction. The test is based on the k n largest observations, where k n is any intermediate sequence of positive integers. Both power of the test and type I error probability are studied for finite sample sizes by simulation. © 2004 Elsevier B.V.All rights reserved. MSC: 62G10; 62G20; 62G32 Keywords: Generalized extreme value and generalized Pareto distributions; Consistency of a test; Semi-parametric approach; Regular variation; Simulation ∗ Corresponding author. Tel.: +351 21 7500414; fax: +351 21 7500081. E-mail addresses: claudia@mat.ua.pt (C. Neves), jan.picek@vslib.cz (J. Picek), isabel.alves@fc.ul.pt (M.I. Fraga Alves). 1 Research partially supported by FCT/POCTI/FEDER. 2 Czech Republic Grant KJB3042303. 0378-3758/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2004.09.008