Statistical Methodology 8 (2011) 172–184 Contents lists available at ScienceDirect Statistical Methodology journal homepage: www.elsevier.com/locate/stamet On model selection in the case of nested distributions — An application to frailty models P. Economou Department of Engineering Science, University of Patras, Rion-Patras, Greece article info Article history: Received 25 October 2009 Received in revised form 4 September 2010 Accepted 4 September 2010 Keywords: Nested distributions Test statistics Diagnostic plot Frailty abstract A number of criteria, test statistics and diagnostic plots have been developed in order to test the adapted distribution assumption. Usually, a simpler distribution is tested against a more complicated one (by adding an extra parameter), which include the first distribution as a special case (nested distributions). In this paper, two new tests are developed in order to test nested distributions. This work concentrates on the case where the extra parameter lies on the boundary of the parameter space. The developed tests are applied on testing the frailty term on frailty models. Simulation results are presented for comparison with other test statistics and the methods are illustrated on two real data sets. © 2010 Elsevier B.V. All rights reserved. 1. Introduction A number of tests have been developed in parametric data analysis in order to determinate if the adapted distribution assumption is the proper one. An interesting case is when the tested distributions are nested, in the sense that one is a special case of the other. There are many examples of such cases, and the most common ones, but not the only ones, are formed by including an extra shift γ parameter in the initial distribution. Bai et al. [1] performed a simulation study using eight different criteria, namely the Akaike’s Information Criterion (AIC), the Schwart’s Information Criterion (SIC), the Kullback–Leibler Information Criterion (KL), a new Generalized Information Criterion (GIC) proposed by them, two criteria related to the upper percentile error (UPE), the entire or Full Percentile Error (FPE), the Chi-square test (CHI) and the Kolmogorov–Smirnov (KS) test, in order to discriminate among two- parameter and three-parameter nested alternative distributions. In their study they have considered three widely used three-parameter (shape–scale–location/shift) distributions, namely the Weibull, the Gamma and the log-normal distributions, against their two-parameter versions in which the location parameter is set equal to zero. E-mail address: peconom@upatras.gr. 1572-3127/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.stamet.2010.09.002