Edgeworth Series Approximation for Chi-square Type Chance Constraints

We introduce two methods for approximation to distribution of weighted sum of chi-square random variables. These methods can be more useful than the known methods in literature to transform chi-square type chance constrained programming (CCP) problem into deterministic problem. Therefore, these are compared with Sengupta (1970)’s method. Some examples are illustrated for the purpose of comparing the solutions of these methods. 1. Introduction The distribution of positive weighted combination of chi-square random variables with any degrees of freedom arises in many application areas such as communication theory, reliability of systems, engineering, industry etc. Many authors are interested in obtaining such as above distribution. Literature review of this subject was given by Johnson et al. (1994). Recent work of Castaño et al. (2005) derived a Laquerre expansion which has been used to evaluate distribution function of the sum of weighted central chi-square random variables. But it is more complicated for using stochastic programming. The best known example of a skewed sum is the chi-square distribution, of course chi-square distribution is itself asymptotically normal, thus this led us to approximate the distribution of a sum of weighted chi-square random variables by an expansion method based on central limit theorem (see Kendall, 1945; Patnaik, 1949; Feller, 1966 and Lehmann, 1999). Next section can be viewed as recalling Castaño et al. (2005)’s work. Introduced methods will be compared with their work which can be seen a main tool of this paper for further discussions. In section three, we will give two methods based on normal approximation. These are called …rst and second edgeworth expansion respectively. We can adapt suggested methods to linear combination of independent random variables assumed to having …nite fourth central moments. Chi-square Received by the editors Sept. 28, 2007; Accepted: Dec. 25, 2007. 2000 Mathematics Subject Classi…cation. Primary 62E20, 65K99; Secondary 60E07, 90C15.