(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)

∂2pθ(x)

∂θ ∂x ≥ ∂pθ(x)

∂θ

∂pθ(x)

∂x for all θ and x.