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 σfinite 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.]