Suppose X = (U, Z), the density of X factors into p,(x) = c(, )g(u; z)h(z)k(u, z),

Suppose X = (U, Z), the density of X factors into pθ,ϑ(x) = c(θ, ϑ)gθ(u; z)hϑ(z)k(u, z),

and the parameters θ, ϑ are unrelated. To see that these assumptions are not enough to insure that Z is S-ancillary for θ, consider the joint density C(θ, ϑ)e − 1 2 (u−θ)2− 1 2 (z−ϑ)2 I(u, z), where I(u, z) is the indicator of the set {(u, z) : u ≤ z}. [Basu (1978).]
Section 10.3