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Simple manipulations show that the latter condition is equivalent to 1 − F G−1 (1 − v) 1 − F G−1 (1 − u) ≤ u v for all 0 < u ≤ v < 1. 8) For discrete random variables that take on values in N the deﬁnition of ≤hr can be written in two diﬀerent ways. Let X and Y be such random variables. We denote X ≤hr Y if P {X = n} P {Y = n} ≥ , P {X ≥ n} P {Y ≥ n} n ∈ N. 9) Equivalently, X ≤hr Y if P {X = n} P {Y = n} ≥ , P {X > n} P {Y > n} n ∈ N. 9) holds if, and only if, P {X ≥ n1 }P {Y ≥ n2 } ≥ P {X ≥ n2 }P {Y ≥ n1 } for all n1 ≤ n2 .

A hazard rate order comparison of random sums is given in the following result. 7. Let {Xi , i = 1, 2, . . } be a sequence of nonnegative IFR independent random variables. 10)), and assume that M and N are independent of the Xi ’s. Then M N Xi ≤hr i=1 Xi . 3(d)) does not have the property of being simply closed under mixtures. However, under quite strong conditions the order ≤hr is closed under mixtures. This is shown in the next theorem. 8. Let X, Y , and Θ be random variables such that [X Θ = θ] ≤hr [Y Θ = θ ] for all θ and θ in the support of Θ.

5) holds. Select an a and a b such that a < b. Then P {u ≤ Y ≤ b} P {u ≤ X ≤ b} ≤ P {a ≤ X ≤ b} P {a ≤ Y ≤ b} whenever u ∈ [a, b]. It follows then that P {a ≤ X < u} P {a ≤ Y < u} ≥ P {u ≤ X ≤ b} P {u ≤ Y ≤ b} whenever u ∈ [a, b]. P {a ≤ X < u} P {u ≤ X ≤ b} ≥ P {a ≤ Y < u} P {u ≤ Y ≤ b} whenever u ∈ [a, b]. C The Likelihood Ratio Order 45 P {b ≤ X ≤ v} P {u ≤ X < b} ≥ . P {u ≤ Y < b} P {b ≤ Y ≤ v} Therefore, when X and Y are continuous random variables, P {a ≤ X < u} P {b ≤ X ≤ v} ≥ P {a ≤ Y < u} P {b ≤ Y ≤ v} whenever a < u ≤ b ≤ v.