Ancillary statistic (${\mathcal{L}}_{\theta }(A)\text{ is independent of }\theta$)

A statistic is ancillary for a statistical model if its probability distribution is the same for every parameter value. Its marginal law carries no information about the parameter, but its observed value can identify the precision or configuration of the experiment. Fisherian conditional inference uses the conditional distribution of an informative statistic given the observed ancillary value. Ancillarity does not mean that the ancillary is independent of every informative statistic.

For independent observations uniform on $(\theta -1/2,\theta +1/2)$, let $U$ and $V$ be the sample minimum and maximum, $R=V-U$, and $C=(U+V)/2$. For $n\ge 2$ their joint density in $(c,r)$ is $n(n-1)r^{n-2}$ on $0<r<1$, $|c-\theta |<(1-r)/2$. Integrating over $c$ gives $f_R(r)=n(n-1)r^{n-2}(1-r)$, independent of $\theta$, so $R$ is an ancillary statistic. Moreover $C\mid R=r$ is uniform on $(\theta -(1-r)/2,\theta +(1-r)/2)$. Consequentlyis an exact conditional $(1-\alpha )$ confidence interval for $\theta$. The realized range determines the appropriate precision, although the range is not itself evidence about the value of $\theta$.

A maximal ancillary statistic has no strictly finer ancillary partition of the sample space, but different maximal ancillary partitions can exist. Let $(U,V)$ take values in $\{0,1{\}}^2$, with probabilities $\theta /2$ on each equal pair and $(1-\theta )/2$ on each unequal pair, for $0<\theta <1$. Both $U$ and $V$ have the Bernoulli distribution with success probability $1/2$ for every $\theta$. Their joint law depends on $\theta$, so combining them is not ancillary. Each partition is maximal: splitting one of its two-point cells singles out a point whose probability depends on $\theta$. There is no common ancillary refinement, and conditioning on one ancillary choice need not specify a unique conditioning principle for all models.

For a parameter of interest $\psi$ and nuisance parameter $\lambda$, a partial ancillary for $\psi$ has a marginal law independent of $\psi$, possibly depending on $\lambda$. This differs from a fully ancillary statistic, whose law is independent of both. For independent Poisson counts with means $\lambda \psi$ and $\lambda (1-\psi )$, their total is Poisson with mean $\lambda$, while the first count conditional on the total is binomial with success probability $\psi$. Conditioning on the total eliminates $\lambda$. A partial ancillary does not automatically eliminate a nuisance parameter; the conditional law must actually be free of that parameter.

For a sufficient statistic $(C,S)$ and parameter $(\psi ,\lambda )$, a useful strong reduction requires the law of $S$ to be free of $\psi$ and the conditional distribution of $C$ given $S$ to be free of the nuisance parameter $\lambda$. In the discrete case it giveswhere ${\varphi }_0$ sums to one on each $(C,S)$ fiber, ${\varphi }_S$ sums to one over $s$, and ${\varphi }_C$ sums to one over $c$ separately for every fixed $s$. Ordinary ancillarity, or just the law of $S$ being free of $\psi$, does not by itself guarantee that conditioning removes $\lambda$.