= Conditional inference in a location-scale family
{title2=$\mathcal L_{\mu,\sigma}(\bar Y,s\mid A=a)$}
For normalized residuals $A_i=(Y_i-\bar Y)/s$, <Fisherian conditional inference> uses the <conditional distribution> of $(\bar Y,s)$ given the observed $A=a$. For almost every attainable ancillary value, the joint conditional density of $\bar Y=m$ and $s=t>0$ is proportional to
$$
t^{n-2}\sigma^{-n}\prod_{i=1}^nf_0\!\left(\frac{m+ta_i-\mu}{\sigma}\right).
$$
The factor $t^{n-2}$ is the radial Jacobian on the $(n-1)$-dimensional centered residual space. After setting $u=(m-\mu)/\sigma$ and $v=t/\sigma$, the conditional density is proportional to $v^{n-2}\prod_i f_0(u+va_i)$, which is parameter-free. Conditioning on a continuous ancillary uses a <regular conditional distribution>, rather than division by the probability of a singleton ancillary value.
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