Invariance of likelihood ratio. Let the family of distributions P = {P, } be dominated

Invariance of likelihood ratio. Let the family of distributions P =

{Pθ, θ ∈ Ω} be dominated by µ, let pθ = dPθ/dµ, let µg−1 be the measure defined by µg−1(A) = µ[g−1(A)], and suppose that µ is absolutely continuous with respect to µg−1 for all g ∈ G.

(i) Then pθ(x) = pgθ¯ (gx) dµ

dµg−1 (gx) (a.e. µ).

(ii) Let Ω and ω be invariant under G¯, and countable. Then the likelihood ratio supΩ pθ(x)/ supω pθ(x) is almost invariant under G.
(iii) Suppose that pθ(x) is continuous in θ for all x, that Ω is a separable pseudometric space, and that Ω and ω are invariant. Then the likelihood ratio is almost invariant under G.