Before revision:  The main theoretical tool is the tilted-Edgeworth technique . It yields a density approximation that is always non-negative, does not need resampling, and is accurate in the tails.
After revision:  The main theoretical tool is the tilted-Edgeworth technique in a non-identically distributed setting. The density approximation is always non-negative, does not need resampling, and is accurate in the tails.