  In this work, we introduce a new acquisition function for sequential sampling
to efficiently quantify rare-event statistics of an input-to-response (ItR)
system with given input probability and expensive function evaluations. Our
acquisition is a generalization of the likelihood-weighted (LW) acquisition
that was initially designed for the same purpose and then extended to many
other applications. The improvement in our acquisition comes from the
generalized form with two additional parameters, by varying which one can
target and address two weaknesses of the original LW acquisition: (1) that the
input space associated with rare-event responses is not sufficiently stressed
in sampling; (2) that the surrogate model (generated from samples) may have
significant deviation from the true ItR function, especially for cases with
complex ItR function and limited number of samples. In addition, we develop a
critical procedure in Monte-Carlo discrete optimization of the acquisition
function, which achieves orders of magnitude acceleration compared to existing
approaches for such type of problems. The superior performance of our new
acquisition to the original LW acquisition is demonstrated in a number of test
cases, including some cases that were designed to show the effectiveness of the
original LW acquisition. We finally apply our method to an engineering example
to quantify the rare-event roll-motion statistics of a ship in a random sea.
