Transformation model ($gY\sim P_{g\theta }\text{when }Y\sim P_{\theta }$)

A group action on the sample space induces a compatible action on the parameter space, preserving the family of probability laws. In a transitive model every law arises by transforming a fixed reference law. A location-scale model is generated by $y\mapsto a+by$, $b>0$, with parameter action $(\mu ,\sigma )\mapsto (a+b\mu ,b\sigma )$.

An estimator respects the group action if the displayed identity holds. If it takes values in the transformation group itself, normalization $T(y{)}^{-1}y$ is a maximal invariant: equivariance proves invariance, and equality of normalized samples reconstructs the transformation between them. For estimates in a homogeneous parameter space with a nontrivial stabilizer, normalization is defined only modulo that stabilizer; quotienting the remaining action is essential.

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.

On samples of distinct real observations, the labeled rank vector is a maximal invariant for simultaneous strictly increasing bijections of the real line. Such transformations preserve every comparison. Conversely two samples with the same ranks can be matched by a piecewise linear increasing bijection through their corresponding ordered observations, with positive-slope tails. Under a common continuous probability distribution, every labeled ordering is equally likely by exchangeability, giving distribution-free rank tests. This argument does not assume that all continuous distributions lie in one orbit of real-line homeomorphisms.

If $T$ is a maximal invariant for a group action, an invariant test function is constant on each group orbit and therefore equals $b\circ T$ for a function $b$ on the orbit space. Conversely every such function is invariant. Appropriate quotient measurability is understood. If the action is transitive on a null family of probability distributions, invariance also gives the same null law of $T$ under every member. Thus tests based on $T$ eliminate that null nuisance variation.