A Bayesian forecasting model: predicting U.S. male mortality by Pedroza C.

By Pedroza C.

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This result is generalized in the following lemma. 6. 24) it be representable in x) + C 2 (b(q», where on b. x* order for to be a solution is sufficient for C(b(q), x) to the form C (b(q), x) = C 1 (b(q), C 1 (b(q), x) is a linear function Proof. By the linearity of C1 (b (q), x) we have EC (b (q), x) E [C 1 (b (q), x) + C:! (b (q»] = C1 (EB (q), x) + EC 2 (b (q», so that min EC (b (q), x) x = min {C 1 (Eb (q), x x) + EC 2 (b (q»). 31) On the other hand, C (Eb (q), x) = C1 (Eb (q), x) + C2 (Eb (q», whereupon min C (Eb (q), x) x = min {CdEb (q), x x) + C2 (Eb (q»).

30) minC(Eb, x), x it is sufficient to have linearity of the function C (b(q), x) on b. Proof. Let C (b (q), x) be a linear function on b. Then by the properties of the expectation EC (b (q), x) = C (Eb, x). From this we infer the assertion of the lemma. 24). This result is generalized in the following lemma. 6. 24) it be representable in x) + C 2 (b(q», where on b. x* order for to be a solution is sufficient for C(b(q), x) to the form C (b(q), x) = C 1 (b(q), C 1 (b(q), x) is a linear function Proof.

Let z=[;J. a(z,a)='f(x)+~(y), a=b, a(a) a(b), = andf (z, a) = g (x) + h (y) - b. 1. Inasmuch as g and h are concave, we have g (AXl + (1 -I,) x~) :> Ag (Xl) + (1 - A) g (X2), h (Ayl + (1 - A) y2) > Ah (yl) + (1 _ A) h (y2), - AbI-(l- A)b 2 = -Abl - (1- A)b 2 • Adding these relations and usingf (z, a) = g (x) + h (y) - b and a = b, we arrive at f (AZl + (1 - A) Z2; -Abl + (1 - A) b2J > Af (Zl, bl )+ (1 - A) f (Z2, b2). The latter implies that f is concave in [ziai]. The convexity of the function () (z, a) in [z i a i ] is proved analogously.

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