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 ∈ } be dominated by μ, let pθ = d Pθ/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 and ω be invariant under G¯ , and countable. Then the likelihood ratio sup pθ(x)/ supω pθ(x) is almost invariant under G.

(iii) Suppose that pθ(x) is continuous in θ for all x, that is a separable pseudometric space, and that and ω are invariant. Then the likelihood ratio is almost invariant under G.