Efficient Importance Sampling for Events of Moderate Deviations with Applications

Abstract

We propose a method to find the alternative distribution in importance sampling. The alternative distribution is optimal in the sense that the asymptotic variance is minimized for estimating tail probabilities of asymptotically normal statistics. Our contribution to importance sampling is three-fold. To begin with, we obtain an explicit expression for the mean of the optimal alternative distribution and the expression motivates a recursive approximation algorithm. Secondly, a new multi-dimensional exponential tilting formula is presented. Lastly, a conservative estimate of the variance is given to facilitate a quick comparison among different stratified sampling schemes in conjunction with importance sampling. Several numerical examples illustrating the efficacy of the proposed method are also included. These results indicate that the proposed method is considerably more efficient than the method based on large deviations theory and the efficiency gain is more significant in higher dimensions. Join work with Dr. Cheng-Der Fuh