Location-scale family ($f(y;\mu ,\sigma )={\sigma }^{-1}{f}_0((y-\mu )/\sigma )$)

In a location-scale family, $Y=\mu +\sigma Z$ for a fixed base probability distribution of $Z$, with $\mu \in \mathbb{R}$ and $\sigma >0$. Location translates observations, and scale multiplies their deviations. For an independent sample with a base density and at least two observations, the normalized residual vector $(Y_i-\bar{Y})/s$, where $s^2=n^{-1}{\sum }_i(Y_i-\bar{Y}{)}^2$, is an ancillary statistic. Indeed, $\bar{Y}=\mu +\sigma \bar{Z}$ and $s=\sigma s_Z$, making the residual vector a function of the base sample alone. This algebraic result needs no population moments.

If $W$ has probability density function $f(w;q)$ and cumulative distribution function $F_W(w;q)$, then $T=\exp (\alpha +\sigma W)$, $\sigma >0$, has density $f((\log t-\alpha )/\sigma ;q)/(\sigma t)$ and distribution function $F_W((\log t-\alpha )/\sigma ;q)$ for $t>0$. The factor $1/(\sigma t)$ follows from the change-of-variables formula for a probability density. Upper truncation at $M$ divides the density by $F_W((\log M-\alpha )/\sigma ;q)$.

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 toThe 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 $v^{n-2}{\prod }_i{f}_0(u+v{a}_i)$, 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.