Before revision:  We provide an algorithm and Monte Carlo experiments illustrating its good performance over first-order asymptotics and Edgeworth expansions , while preserving analytical tractability .
After revision:  We provide an algorithm and Monte Carlo experiments on density approximation and testing in the presence of nuisance parameters illustrate the good performance of our approximation over first-order asymptotics and Edgeworth expansions , while preserving analytical tractability .