= Saddlepoint density approximation
{title2=$\widehat f_{S_n}(nx)=\frac{e^{n[K(\widehat t)-\widehat t x]}}{\sqrt{2\pi nK''(\widehat t)}}$}
= Statistical saddlepoint approximation
{synonym}
For independent identically distributed continuous random variables, let $K(t)=\log\mathbb E e^{tY}$ be finite near zero. At a mean value $x$ reached by $K'(\widehat t)=x$ in the interior of that domain, the displayed leading approximation describes the density of the sum $S_n$. <Exponential tilting> by $\widehat t$ puts the sum at its tilted mean; a local normal approximation gives the denominator, and undoing the tilt gives the exponential factor. Under suitable smooth nonlattice regularity, at fixed interior $x$ the exact density is the displayed quantity times $1+O(n^{-1})$. This statistical approximation is an application of the <saddle-point approximation> to a transform inversion integral. It requires more than the existence of a few ordinary moments.
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