Partial ancillary statistic (${\mathcal{L}}_{\psi ,\lambda }(A)={\mathcal{L}}_{\lambda }(A)$)

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 $\lambda$.