= Solution
In <statistical decision theory>, the <statistical invariance principle> says that a procedure should respect transformations that leave the inferential problem unchanged. If a <group action> preserves the model and carries each of the null and alternative parameter sets into itself, an invariant test has rejection probability function $\varphi(gy)=\varphi(y)$. A <maximal invariant> identifies its orbits, so every invariant test factors through that statistic: define its value at an orbit using any representative; invariance makes the definition <independent> of the representative. If the <group> is transitive on the null parameter set, the null law of any invariant statistic is free of that <nuisance parameter>.
For example, testing a normal mean $\mu=\mu_0$ with unknown scale, positive transformations $y_i-\mu_0\mapsto b(y_i-\mu_0)$ preserve a one-sided mean alternative. The statistic $\sqrt n(\bar Y-\mu_0)/s$, with the usual unbiased sample-<variance> scale $s$, is invariant and has the <Student t-distribution> with $n-1$ degrees of freedom under the null. For estimation the corresponding requirement is an <equivariant estimator>, together with a <loss function> invariant under simultaneous transformation of decision and parameter. Invariance alone does not prove unrestricted optimality or supply a unique best procedure.
There is a distinct probabilistic use of the term: the <Donsker invariance principle> states that centred iid partial sums, normalized by $\sigma\sqrt n$, converge as processes on $[0,1]$ to <Brownian motion>. Its empirical-process version gives a <Brownian bridge>, used by goodness-of-fit tests in the next note. These functional limit theorems should be distinguished from the <group>-based inferential principle.
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