= Lugannani-Rice saddlepoint tail approximation
{c}
For a sample mean from an independent identically distributed sample, let $K(t)$ be the <cumulant-generating function> of one observation and solve $K'(t)=x$. Put $w=\operatorname{sgn}(t)\sqrt{2n(tx-K(t))}$ and $u=t\sqrt{nK''(t)}$. Under the smooth, nonlattice conditions for a <saddlepoint density approximation>, the corresponding approximation to the <cumulative distribution function> is
$$
\Pr(\overline X\le x)\simeq\Phi(w)+\phi(w)\left(\frac1w-\frac1u\right),
$$
where $\Phi$ and $\phi$ are the standard <normal distribution> functions. The apparent singularity at $t=0$ is removable. Write $v=K''(0)$ and $k=K'''(0)$; <Taylor expansion> gives $w=t\sqrt{nv}(1+kt/(3v)+O(t^2))$ and $u=t\sqrt{nv}(1+kt/(2v)+O(t^2))$. Thus $1/w-1/u\to k/(6\sqrt n\,v^{3/2})$. The approximation requires suitable tail regularity and is not a universal formula for distributions without a moment-generating function.
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