Let F and G be two continuous, strictly increasing c.d.f.s, and let k(u) = G[F−1(u)], 0 < u < 1.

(i) Show F and G are stochastically ordered, say F(x) ≤ G(x) for all x, if and only if k(u) ≤ u for all 0 < u < 1.

(ii) If F and G have densities f and g, then show they are monotone likelihood ratio ordered, say g/ f nondecreasing, if and only if k is convex.

(iii) Use (i) and (ii) to give an alternative proof of the fact that MLR implies stochastic ordering.