Maximal invariant

An invariant statistic that takes different values on the different orbits generated by a group of one-to-one measurable transformations of the sampling space. Thus, if $(\mathfrak X,\mathfrak B)$ is a sampling space and $G=\{g\}$ is a group of one-to-one $\mathfrak B$-measurable transformations of $\mathfrak X$ onto itself, then an invariant statistic $T(x)$ is a maximal invariant if $T(x_2)=T(x_1)$ implies that $x_2=gx_1$ for some $g\in G$. For example, if $\mathfrak X=\mathbf R^n$, $x=(x_1,\ldots,x_2)^T$, $G=\{\Gamma\}$ is the group of orthogonal transformations $\mathbf R^n\to\mathbf R^n$, and $y=\Gamma x$, then the statistic $T(x)=\sum x_i^2$ is a maximal invariant. Any invariant statistic is a function of the maximal invariant.

Maximal invariants are used for the construction of invariant tests (cf. Invariant test).

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
[2]	S. Zacks, "The theory of statistical inference" , Wiley (1975)
[3]	G.P. Klimov, "Invariant inferences in statistics" , Moscow (1973) (In Russian)