(i) A necessary and sufficient condition for densities p(x) to have monotone likelihood ratio in x, if...

(i) A necessary and sufficient condition for densities pθ(x) to have monotone likelihood ratio in x, if the mixed second derivative ∂2 log pθ(x)/∂θ ∂x exists, is that this derivative is ≥ 0 for all θ and x.

(ii) An equivalent condition is that pθ(x)
∂2 pθ(x)
∂θ ∂x ≥
∂ pθ(x)
∂θ
∂ pθ(x)
∂x for all θ and x.