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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.

In a parametric statistical model $\{P_{\theta }:\theta \in \Theta \}$, the likelihood function $L(\theta ;y)$ measures the relative support given by the observed sample for different parameter values. Maximum likelihood estimation supplies a point estimate; the sampling distributions of suitable statistics calibrate significance tests and confidence intervals. A sufficient statistic retains all parameter-dependent information in the likelihood function, so one first seeks a reduction to such a statistic rather than discarding information arbitrarily.

The characteristic conditioning principle in Fisherian conditional inference is to calibrate inference conditional on the observed value of an ancillary statistic. With no nuisance parameter, a statistic $A$ is ancillary when its law is the same for every $\theta$. Its observed value can nevertheless describe the experimental configuration or precision. Conditional inference uses ${\mathcal{L}}_{\theta }(T\mid A=a)$ for an informative statistic $T$, with $a$ the observed ancillary value. Since the marginal law of $A$ does not depend on $\theta$, conditioning leaves the parameter-dependent likelihood factor unchanged, while potentially changing the appropriate repeated-sampling reference distribution. Conditioning on a continuous $A$ means using a regular conditional distribution, rather than dividing by $\mathbb{P}(A=a)$.

With parameters $(\psi ,\lambda )$, where $\psi$ is the target and $\lambda$ a nuisance parameter, a fully ancillary statistic has a law independent of both. A partial ancillary statistic for $\psi$ may have a law depending on $\lambda$ but not on $\psi$. To eliminate $\lambda$ by conditioning, one needs the actual conditional law of the informative statistic to be free of $\lambda$; partial ancillarity by itself does not guarantee this. For example, if independent counts have Poisson distributions with means $\lambda \psi$ and $\lambda (1-\psi )$, their total $N$ has Poisson distribution with mean $\lambda$, independent of $\psi$. The first count conditional on $N=m$ has binomial distribution with parameters $(m,\psi )$. Conditioning removes the nuisance intensity $\lambda$ exactly. More generally, nuisance-free pivotal quantities or conditional likelihoods can be used where available; there is no universal conditioning device that removes every nuisance parameter.

For a concrete example of non-uniqueness of maximal ancillary statistics, consider one observed pair $(U,V)\in \{0,1{\}}^2$, withBoth $U$ and $V$ have Bernoulli distribution with success probability $1/2$, independent of $\theta$, so each is an ancillary statistic. Their generated partitions are different and incomparable. Moreover, each is maximal: refining either two-point cell singles out a point whose probability is $\theta /2$ or $(1-\theta )/2$, which is parameter-dependent. Their joint statistic is not ancillary, since $P_{\theta }(U=V)=\theta$. Thus there is no common finer ancillary partition that reconciles the two choices. The conditional laws differ as labelled experiments: $P_{\theta }(V=U\mid U)=\theta$ and $P_{\theta }(U=V\mid V)=\theta$, but they condition on different observed information. This illustrates why “condition on a maximal ancillary” need not specify a unique procedure.

For the location-scale family, write $Y_i=\mu +\sigma Z_i$, where the $Z_i$ are independent with the fixed density $f_0$. Let $s_Z^2=n^{-1}{\sum }_i(Z_i-\bar{Z}{)}^2$. The transformation givesand thereforeFor $n\ge 2$, $s_Z>0$ almost surely, because an independent sample from a density has probability zero of all observations coinciding. The right side depends only on a sample from the fixed base law, so the entire vector $A$ has a law independent of $(\mu ,\sigma )$: $\boxed{A\text{ is ancillary}.}$ No finite population mean or variance is needed for this algebraic argument. With one observation the displayed standardization is undefined, so at least two observations are implicitly required.

The pair $(\hat{\mu },\hat{\sigma })$ together with $A$ reconstructs the full sample through $Y_i=\hat{\mu }+\hat{\sigma }{A}_i$. Thus conditional inference in a location-scale family useswhere $a$ is the observed residual configuration. Equivalently one can use the parameter-free conditional law of $((\hat{\mu }-\mu )/\sigma ,\hat{\sigma }/\sigma )$ given $A=a$. To see this explicitly, the centered residual subspace has dimension $n-1$, so its radial coordinate $t=\hat{\sigma }$ contributes a Jacobian determinant factor proportional to $t^{n-2}$. The conditional density of $\hat{\mu }=m$, $\hat{\sigma }=t>0$ is therefore proportional toSetting $u=(m-\mu )/\sigma$ and $v=t/\sigma$ gives a conditional density proportional to $v^{n-2}{\prod }_i{f}_0(u+v{a}_i)$, with no unknown parameter. The normalizing factor can depend on $a$, but not on $\mu$ or $\sigma$. These statements concern almost every attainable ancillary value.