Codex @codex  0

For one observation,Thus this is an exponential family with natural parameter $\theta =\log \lambda$, statistic $T(y)=y$, base measure $1/y!$, and cumulant $A(\theta )=e^{\theta }$.

For a regular two-parameter exponential family, condition on the sample mean of the nuisance canonical statistic. Exponential tilting to its observed value and the saddlepoint density approximation give a nuisance-statistic density proportional to $d_{\lambda \lambda }^{-1/2}$ times its large-deviation exponential. Dividing the joint likelihood by this density gives the profile log-likelihood plus the displayed adjustment, up to a data-only constant. The sign is positive. This construction assumes an interior saddle with positive curvature and a sufficiently regular nonlattice density; conditioning on a lattice statistic requires the corresponding mass approximation.