 Solution

A conditional likelihood is the conditional density of the observed data given a chosen statistic, viewed as a function of the parameter. Conditioning can eliminate a nuisance parameter if the statistic is sufficient for it within the relevant family. The conditioning statistic need not be ancillary; conditioning can also discard some information about the parameter of interest.

For the canonical family, let $T_1={\sum }_i{\tau }_1(Y_i)$ and $s=n^{-1}{\sum }_i{\tau }_2(Y_i)$. Normalization of the density gives the cumulant-generating function of one nuisance statistic:For the sample mean $S$, it is $K_S(t)=n[d(\psi ,\lambda +t/n)-d(\psi ,\lambda )]$. Differentiation gives $d_{\lambda }=\mathbb{E}{\tau }_2(Y_i)$ and $d_{\lambda \lambda }=\mathrm{Var}({\tau }_2(Y_i))$.

Assume an interior saddle with positive curvature and the usual smooth nonlattice density conditions. Define ${\hat{\lambda }}_{\psi }$ byThis is the constrained maximum-likelihood estimator of $\lambda$ with $\psi$ held fixed. The exponential tilt parameter for one summand is $\hat{t}={\hat{\lambda }}_{\psi }-\lambda$. Tilting puts the sample mean at $s$, where its tilted density has leading normal value $\sqrt{n/[2\pi d_{\lambda \lambda }(\psi ,{\hat{\lambda }}_{\psi })]}$. Undoing the tilt gives the saddlepoint density approximationThis is the density of the mean, not the sum; their factors differ by $n$.

The sample log-likelihood is $\ell (\psi ,\lambda )=\psi T_1+n\lambda s-nd(\psi ,\lambda )-{\sum }_{i}Q(Y_i)$. Conditional on $S=s$, the factor involving $\lambda$ is constant on the conditioning surface, so the conditional distribution is free of $\lambda$. Subtracting the saddlepoint log-density from the joint log-likelihood now cancels all the original nuisance terms:The final term, and any conditioning-coordinate factor, are data-only constants. Thus the saddlepoint conditional likelihood adjustment isThe absolute-value notation agrees with the question; positive variance makes it unnecessary in this regular setting. The sign of this correction is positive, because a nuisance-statistic density with inverse-square-root curvature has been divided out. A lattice-valued statistic requires the corresponding mass approximation, and a boundary or zero-variance saddle is outside the stated regular density calculation.