A reflection-based variance reduction technique for sum of random variables

Monte Carlo simulation has been widely used as a standard tool for estimating expectations. In this paper we develop a variance reduction technique for a particular case when the expectation is taken under a constraint that a sum of random variables is larger than a threshold. The proposed technique is based on a reflection argument on the sample space and requires knowing the joint density of the random variables. It turns out the technique can always guarantee a variance reduction. More importantly, the technique sheds light on how observations violating the constraint can be used more efficiently in estimation, compared to crude Monte Carlo.