Efficient importance sampling for large sums of independent and identically distributed random variables

Abstract We discuss estimating the probability that sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $$\mathbb {P}(\sum _{i=1}^{N}{X_i} \le \gamma )$$ P ( ∑ i = 1 N X ≤ γ ) , via importance sampling (IS). are particularly interested in rare event regime when N is large and/or $$\gamma $$ small. The exponential twisting popular technique for similar problems that, most cases, compares favorably to other estimators. However, it has some limitations: (i) It assumes knowledge moment-generating function $$X_i$$ (ii) under new IS PDF not straightforward might be expensive. aim this work propose an alternative approximately yields, certain classes distributions regime, at least same performance as and, time, does introduce serious limitations. first class includes whose density functions (PDFs) asymptotically equivalent, $$x \rightarrow 0$$ x → 0 $$bx^{p}$$ b p $$p>-1$$ > - $$b>0$$ . For distributions, Gamma with appropriately chosen parameters retrieves approximately, corresponding small values estimator based on use technique. In second class, we consider Log-normal setting, zero vanishes faster than any polynomial, show numerically optimized clearly outperforms PDF. Numerical experiments validate efficiency proposed delivering highly accurate estimate