= Solution
Let a <group> $G$ act on the sample space. A statistic $T$ is a <maximal invariant> when it is invariant and separates <group orbits>:
$$
T(gx)=T(x),\qquad
T(x)=T(y)\ \Longrightarrow\ y=gx\text{ for some }g\in G.
$$
Together these conditions say that $T(x)=T(y)$ exactly when $x$ and $y$ belong to the same orbit.
The basic result is that <invariant tests factor through a maximal invariant>. Specifically, a test function $\varphi(x)\in[0,1]$ is invariant exactly when it can be written $b(T(x))$. To prove the nontrivial direction, define $b(t)=\varphi(x)$ for any representative with $T(x)=t$. Two representatives are in the same orbit by maximality, and invariance gives them the same test value, so $b$ is well-defined. Conversely $b(T(gx))=b(T(x))$ proves invariance. Measurability is taken on the quotient statistic space; for the finite rank spaces used below it is automatic.
If a null family is transitive under the <group action>, the null law of $T$ is parameter-free. Indeed, if $P_{\theta'}$ is the law of $gX$ when $X\sim P_\theta$, then for every measurable set $B$,
$$
P_{\theta'}\{T\in B\}=P_\theta\{T(gX)\in B\}
=P_\theta\{T(X)\in B\}.
$$
Thus invariant procedures use exactly the information in $T$, and its common null law calibrates tests without estimating the transformed nuisance distribution. This explains the role of maximal invariants in <nonparametric statistics>.
For simultaneous strictly increasing bijections of the real line, the labeled <ranks of observations> are a <maximal invariant> on samples without ties. Increasing maps preserve every comparison. Conversely, two such samples with the same ranks can be matched by a piecewise linear increasing bijection through their corresponding ordered observations, extended with positive-slope tails. This proves <ranks as a maximal invariant under increasing transformations>. Under independent sampling from any common continuous distribution, every labeled ordering is equally likely by <exchangeability>. Hence <rank tests> have distribution-free null calibration even without a global transitivity assertion for the entire class of continuous distributions.
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