Let x = (x1,...,xn), and let g(x, ) be a family of probability densities depending on

Let x = (x1,...,xn), and let gθ(x, ξ) be a family of probability densities depending on θ = (θ1,...,θr) and the real parameter ξ, and jointly measurable in x and ξ. For each θ, let hθ(ξ) be a probability density with respect to a σ-finite measure ν such that pθ(x) = gθ(x, ξ)hθ(ξ) dν(ξ) exists. We shall say that a function f of two arguments u = (u1,...,ur), v = (v1,...,vs) is nondecreasing in (u, v) if f(u

, v)/f(u, v) ≤ f(u

, v

)/f(u, v

) for all (u, v) satisfying ui ≤ u

i, vj ≤ v

j (i = 1,...,r; j = 1,...,s). Then pθ(x) is nondecreasing in (x, θ)

provided the product gθ(x, ξ)hθ(ξ) is

(a) nondecreasing in (x, θ) for each fixed ξ;

(b) nondecreasing in (θ, ξ) for each fixed x;

(c) nondecreasing in (x, ξ) for each fixed θ.
[Interpreting gθ(x, ξ) as the conditional density of x given ξ, and hθ(ξ) as the a priori density of ξ, let ρ(ξ) denote the a posteriori density of ξ given x, and let ρ

(ξ) be defined analogously with θ
in place of θ. That pθ(x) is nondecreasing in its two arguments is equivalent to  gθ(x

, ξ)
gθ(x, ξ) ρ(ξ) dν(ξ) ≤
 gθ
(x

, ξ)
gθ
(x, ξ) ρ

(ξ) dν(ξ).
By

(a) it is enough to prove that D =  gθ(x

, ξ)
gθ(x, ξ) [ρ

(ξ) − ρ(ξ)] dν(ξ) ≥ 0.
Let S− = {ξ : ρ

(ξ)/ρ(ξ) < 1} and S+ = {ξ : ρ(ξ)/ρ(ξ) ≥ 1}. By

(b) the set S−
lies entirely to the left of S+. It follows from

(c) that there exists a ≤ b such that D = a 
S−
[ρ

(ξ) − ρ(ξ)] dν(ξ) + b 
S+
[ρ

(ξ) − ρ(ξ)] dν(ξ), and hence that D = (b − a)
S+ [ρ

(ξ) − ρ(ξ)] dν(ξ) ≥ 0.]