A comparison of efficient approximations for a weighted sum of chi-squared random variables

Inmany applications, the cumulative distribution function (cdf) FQN of a positively weighted sum of N i.i.d. chi-squared randomvariables QN is required.Although there is no known closed-form solution for FQN , there are many good approximations. When computational efficiency is not an issue, Imhof’smethod provides a good solution. However, when both the accuracy of the approximation and the speed of its computation are a concern, there is no clear preferred choice. Previous comparisons between approximatemethods could be considered insufficient. Furthermore, in streaming data applications where the computation needs to be both sequential and efficient, only a few of the available methods may be suitable. Streaming data problems are becoming ubiquitous and provide the motivation for this paper. We develop a framework to enable a much more extensive comparison between approximate methods for computing the cdf of weighted sums of an arbitrary random variable. Utilising this framework, a new and comprehensive analysis of four efficient approximate methods for computing FQN is performed. This analysis procedure is much more thorough and statistically valid than previous approaches described in the literature. A surprising result of this analysis is that the accuracy of these approximate methods increases with N . Electronic supplementary material The online version of this article (doi:10.1007/s11222-015-9583-4) contains supplementary material, which is available to authorized users. B Dean A. Bodenham [email protected] 1 Department of Mathematics, Imperial College London, London, UK 2 D-BSSE, ETH Zürich, Basel, Switzerland 3 Heilbronn Institute of Mathematics, University of Bristol, Bristol, UK