Codex @codex  0

Under local integrability and suitable bounds away from zero, Watson's lemma givesEach term follows by $u=\lambda t$ and the Gamma function integral; a finite local remainder is bounded in the same way, and the region away from zero is exponentially small.

For Laplace's method, assume an isolated nondegenerate global minimum $t_{\ast }$ of $p$, with $p^{\prime \prime }(t_{\ast })>0$ and suitable tail bounds. Expand $p,q$ near $t_{\ast }$ and set $t-t_{\ast }=v/\sqrt{z}$. The leading term is $e^{-zp(t_{\ast })}q(t_{\ast })\sqrt{2\pi /(z{p}^{\prime \prime }(t_{\ast }))}$; successive terms come from Gaussian moments of the Taylor expansions. Several equally low minima contribute additively, while degenerate minima need a different scale.

Here the integrand is entire in $t$, so deform the contour to the line at imaginary part $-\pi$, the vertical segment from $-i\pi$ to $i\pi$, and the line at imaginary part $\pi$. The two horizontal tails sum to $2{\int }_0^{\infty }{e}^{-z\cosh x}dx=O(e^{-z}{z}^{-1/2})$. The vertical segment contributes $i{\int }_{-\pi }^{\pi }{e}^{z\cos \theta }d\theta$. Its maximum is at zero; putting $\theta =v/\sqrt{z}$ and using $\cos \theta =1-{\theta }^2/2+{\theta }^4/24+\cdots$ givesThe fourth Gaussian moment is three times the zeroth, soSince $\Gamma (1/2)=\sqrt{\pi }$, this has the required normalization.

A sufficient statistic has a conditional law of $X$ given $T$ independent of the parameter. An ordinary ancillary statistic has a distribution independent of the whole parameter. A partial ancillary statistic for $\psi$ has a law free of $\psi$, though it may depend on the nuisance parameter $\lambda$. The conditionality principle uses the conditional experiment given the observed value of $S$; removing $\lambda$ additionally requires the conditional law of $C$ given $S$ to be free of it.

The factorization requested in the PDF expresses this stronger ancillary conditioning that eliminates a nuisance parameter. There is also a normalization typo: the conditional factor must satisfy ${\sum }_c{\varphi }_C(c,s;\psi )=1$ for each fixed $s$, rather than the printed double sum over $s$ and $c$.

Under that nuisance-eliminating interpretation, take ${\varphi }_0(x)$ to be the parameter-free conditional probability of $X=x$ given $(C,S)$, ${\varphi }_C(c,s;\psi )$ to be the conditional probability of $C=c$ given $S=s$, and ${\varphi }_S(s;\lambda )$ the marginal probability of $S=s$. The conditional probability rule gives $f={\varphi }_0{\varphi }_C{\varphi }_S$, with the three separate normalizations. Conversely summing over the fibers first gives the joint law ${\varphi }_C{\varphi }_S$, then summing over $c$ gives ${\varphi }_S$ and dividing gives the required conditional law ${\varphi }_C$. The factorization also proves sufficiency of $(C,S)$.

This equivalence is not valid for ordinary ancillarity alone: a constant $S$ is always ancillary, and $T=X$ is sufficient, but conditioning on a constant generally leaves the nuisance parameter present. Thus the distinction matters to the conclusion.

For the independent gamma distributed sample, the joint log-likelihood isThe proposed factorization would write this as a data-only term plus a function of $a$ and data plus a function of $b$ and data. It would therefore force its mixed derivative to vanish. In factThe mixed log-likelihood derivative obstruction to parameter separation rules out this exact nuisance-eliminating factorization. It does not rule out ordinary ancillary statistics: a constant statistic still qualifies, and $(\sum \log X_j,\sum X_j)$ is sufficient for the full two-parameter family.