A Bayesian Alternative to Parametric Hypothesis Testing by Rueda R.

By Rueda R.

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By an appropriate transformation we can, with no loss in generality, specify that (10) v(P) = E[E(u,P(s)),P*] for all P £ 3C0. According to (8) the proof of Theorem 4 can be completed by proving that (10) holds for all bounded horse lotteries. Our first step in this direction will be to prove that if P (A) = 1 and if c and d defined in the following expression are finite then (11) c = inf {£(«, P(s))\seA\ g v(P) S sup {E{u, P{s))\seA\ = d. Let Q = P on A and t g Q(s) £ d o n i ' . Since A" is null, Q ~ P and hence v (P) = w ( 0 ) by (8).

_„+i Ai. Letting v on 3C be as given in (8) and (9) and satisfying (10) on 3Co, (24) » ( 0 . ) = E ; . ) = P*(A„) _ 1 £ ? _ , p * ( 4 . ) - n for n = 1, 2, • • • since Qn £ 3Co for each n. „(s)) = iP*(Ai)-1 = §P*(A„)~ + \[P*(A,)'1 a for all P*(Ai)-i] - S£ UTA, and, by (21) and (23), E(u, iP(s) 1. <>„(«)) £ i P * ( A „ r ' EXPECTED UTILITY THEORY for all se\J"+iAi. 19 1428 PETER C. FISHBURN Therefore and hence, by (11), v(\P + § O J 5: iP*(A„)~\ implies that »(/>) S P*(A„r1 - P*(A„rl which, using (9) and (24) 2:UP*(A<) + n £ re for re — 1, 2, • • • .

They are illustrated in Exercises CR-4 and ME-2. Further restrictions on the utility function result if one makes additional assumptions concerning the decisionmaker's behavior. For example, an investor who never prefers a fair gamble to the status quo must have a concave utility function (Exercise CR-1). Such investors will not simultaneously gamble and purchase insurance. 1 If the investor's preferences for horse lotteries are independent of his initial wealth level, then the utility function must be linear or exponential (Exercise CR-5).

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