This paper generalizes some results of Wijsman concerning the calculation of the density of a maximal invariant. The idea of the technique is to represent the sample space as a product space, one factor $Z$ being a global cross section, i.e., essentially a set that intersects each orbit in a unique point, and the other factor being a coset space of the invariance group. Integration over the invariance group then gives the distribution of the identity function on $Z$ which is a maximal invariant. Part I of the paper gives sufficient conditions for the technique to be applicable, while Part II exhibits the technique along with an example. Part II is on a more elementary level than Part I and may be understood without a reading of Part I.
Uwe Koehn. "Global Cross Sections and the Densities of Maximal Invariants." Ann. Math. Statist. 41 (6) 2045 - 2056, December, 1970. https://doi.org/10.1214/aoms/1177696704