Let f (x)/[1 − F(x)] be the “mortality” of a subject at time x given that it has survived to this time. A c.d.f. F is said to be smaller than G in the hazard ordering if g(x)

1 − G(x) ≤ f (x)

1 − F(x)

for all x . (3.48)

(i) Show that (3.48) is equivalent to 1 − F(x)

1 − G(x) is nonincreasing.

(ii) Show that (3.48) holds if and only if k is starshaped. [A function k defined on an interval I ⊆ [0,∞) is starshaped on I if k(λx) ≤ λk(x) whenever x ∈ I, λx ∈ I, 0 ≤ λ ≤ 1. Problems 3.43 and 3.44 are based on Lehmann and Rojo (1992).]