  We aim to estimate the probability that the sum of nonnegative independent
and identically distributed random variables falls below a given threshold,
i.e., $\mathbb{P}(\sum_{i=1}^{N}{X_i} \leq \gamma)$, via importance sampling
(IS). We are particularly interested in the rare event regime when $N$ is large
and/or $\gamma$ is small. The exponential twisting is a popular technique that,
in most of the cases, compares favorably to existing estimators. However, it
has several limitations: i) it assumes the knowledge of the moment generating
function of $X_i$ and ii) sampling under the new measure is not straightforward
and might be expensive. The aim of this work is to propose an alternative
change of measure that yields, in the rare event regime corresponding to large
$N$ and/or small $\gamma$, at least the same performance as the exponential
twisting technique and, at the same time, does not introduce serious
limitations. For distributions whose probability density functions (PDFs) are
$\mathcal{O}(x^{d})$, as $x \rightarrow 0$ and $d>-1$, we prove that the Gamma
IS PDF with appropriately chosen parameters retrieves asymptotically, in the
rare event regime, the same performance of the estimator based on the use of
the exponential twisting technique. Moreover, in the Log-normal setting, where
the PDF at zero vanishes faster than any polynomial, we numerically show that a
Gamma IS PDF with optimized parameters clearly outperforms the exponential
twisting change of measure. Numerical experiments validate the efficiency of
the proposed estimator in delivering a highly accurate estimate in the regime
of large $N$ and/or small $\gamma$.
