Maximal invariant
= Maximal invariant
{title2=$A(y)=A(z)\iff z=gy\text{ for some }g$}
A statistic is invariant if $A(gy)=A(y)$; it is maximal when it also separates different <group orbits>. Consequently every invariant procedure factors through the <maximal invariant>, subject to the usual measurability conditions on the orbit space. Standardizing a sample by an <equivariant estimator> of location and positive scale gives a <maximal invariant> for the positive affine <group>.