= Location-scale family
{title2=$f(y;\mu,\sigma)=\sigma^{-1}f_0((y-\mu)/\sigma)$}
= Location-scale model
{synonym}
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.
Back to article page
