Insurance: Mathematics and Economics 50 (2012) 385–390 Contents lists available at SciVerse ScienceDirect Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime Comparison of increasing directionally convex transformations of random vectors with a common copula Félix Belzunce a , Alfonso Suárez-Llorens b , Miguel A. Sordo b,∗ a Dpto. Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain b Dpto. Estadística e Investigación Operativa, Facultad de Ciencias Empresariales, Universidad de Cádiz, C/Duque de Nájera, 8, 11002 Cádiz, Spain article info Article history: Received June 2011 Received in revised form January 2012 Accepted 1 February 2012 MSC: IM30 Keywords: Increasing directionally convex functions Convex order Dispersive order Copula Comparison of variances Premium principles Multivariate distortions abstract Let X and Y be two random vectors in R n sharing the same dependence structure, that is, with a common copula. As many authors have pointed out, results of the following form are of interest: under which conditions, the stochastic comparison of the marginals of X and Y is a sufficient condition for the comparison of the expected values for some transformations of these random vectors? Assuming that the components are ordered in the univariate dispersive order – which can be interpreted as a multivariate dispersion ordering between the vectors – the main purpose of this work is to show that a weak positive dependence property, such as the positive association property, is enough for the comparison of the variance of any increasing directionally convex transformation of the vectors. Some applications in premium principles, optimization and multivariate distortions are described. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Let X and Y be two random vectors in R n sharing the same dependence structure, that is, with the same copula. In this situation, Müller and Scarsini (2001) pointed out the following problem: under which conditions, the stochastic comparison of the marginals of X and Y is a sufficient condition for the comparison of expected values for some transformations of the two vectors? This problem has a clear motivation in the comparison of portfolios in terms of risks, where the individual random returns are transformed by positive linear combinations, specially in situations where we know the dependence structure of the random vector associated to the portfolio but only partial information is available for its marginals. If we know, for example, that the marginals are stochastically ordered with respect to some parametric model of distribution, we can use this ordering to compare the vector of unknown marginals with the vector that we obtain by replacing the unknown marginals by the parametric model. In Risk Theory, another interesting example of random vectors sharing a common ∗ Corresponding author. E-mail addresses: belzunce@um.es (F. Belzunce), alfonso.suarez@uca.es (A. Suárez-Llorens), mangel.sordo@uca.es (M.A. Sordo). copula is given by comonotone random vectors; see Puccetti and Scarsini (2010) for a recent work in this area. Finally, we also find applications of this type of results in reliability. For instance, Navarro and Spizzichino (2010) studied comparisons of series and parallel systems with vectors of component lifetimes sharing the same copula. Let us make a review of some of these results. For some particular marginal orderings, the previous problem has been dealt successfully in the literature. For example, Scarsini (1988) proved that the usual univariate stochastic order between the marginals of two random vectors sharing the same copula implies the usual multivariate stochastic order between the vectors. Scarsini (1998) showed that this result does not hold for the case of the convex order and convex transformations. This led to Müller and Scarsini (2001) to consider some variations to provide a result for some kind of convexity. In particular they considered random vectors where the common copula has a positive dependence structure (conditionally increasing) and they considered comparisons of the two random vectors in terms of expected values of directionally convex transformations. Note that directionally convex transformations include the linear combinations of the components as a particular case. Recently, Balakrishnan et al. (2011) have extended this result to the case of increasing directionally convex transformations and have provided some applications in the context of generalized order statistics. 0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.02.001