Lower convex order bound approximations for sums of log‐skew normal random variables

When it comes to modeling dependent random variables, not surprisingly, the multivariate normal distribution has received the most attention because of its many appealing properties. However, when it comes to practical implementation, the same family of distribution is often rejected for modeling financial and insurance data because they do not apparently behave in the multivariate normal sense. In this paper, we consider the construction of lower convex order bounds, in the sense of Kaas et al. (Insur. Math. Econ. 2000; 27:151–168), to approximate sums of dependent log‐skew normal random variables. The dependence structure of these random variables is based on the class of multivariate closed skew‐normal (CSN) distribution that appears in González‐Farías et al. (Skew‐Elliptical Distributions and their Applications. A Journey Beyond Normality, Genton MG (ed.), Chapter 6. Chapman & Hall/CRC: Boca Raton, FL, 2004; 25–42) and which carries several interesting properties of the normal distribution apart from allowing additional parameters to regulate skewness. The bounds that we present in this paper are therefore natural extensions to the results presented in Dhaene et al. (Insurance: Mathematics and Economics 2002; 31(2): 133–161; Insurance: Mathematics and Economics 2002b; 31(1):3–33), where bounds for sums of log‐normal random variables have been derived. These lower bound approximations are constructed based on the additional information provided by a conditioning variable which when optimally chosen can provide an accurate approximation. We exploit inherent properties of this family of skew‐normal distributions in order to choose the optimal conditioning variable. Results of our simulations provide an indication of the performance of these approximations. Copyright © 2010 John Wiley & Sons, Ltd.