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Lecture19-2004

# Lecture19-2004 - Lecture XIX Increasing Risk and Ranking...

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1 Lecture XIX Increasing Risk and Ranking Functions I. Implications of Definitions A. Theorem 1. Consider cumulative distributions ( ) Fx and () Gx , then there is an increasing continuous function ( ) rx such that: [] Fx d y y () () , [,] , −≥ 0 00 1 if and only if is at least as risky as ( ) by Definition 1. 1. Proof: Assume there exists an increasing twice differentiable ( ) such that y y , . 0 1 Let zrx = and define functions ( ) * . G and * . F by () ( ) *1 Gz G r z = and ( ) ( ) ( ) Fz F = . Then, Grx Frx d y Gz Fzd z y y r ry ** (() ) ) () , , 0 0 1 1 or We know by previous proof (here attributed to Hadar and Russell) that Gz Fzuzd z u z r r () ' ⇒− 0 0 0 0 A couple of notes on this point: a. If you made the assumption that ( ) is a one-to-one mapping, we have simply changed the definition of the cumulative distribution, defining it on the variable z which is related to the original variable x by ( ) = . This assumption would be implied by the imposition ( ) 1 x rz = . b. Along the same lines, we really haven’t changed the bounds of integration. We have only mapped them into the variable z . 2. Mapping the transformation back to x by ( )( ) ( ) vx urx = , the integral becomes

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AEB 6182–Agricultural Risk Analysis and Decision Making Fall 2004 Professor Charles Moss Lecture 19 2 [] Gx Fx d vx () () () −≥ 0 1 0 Integrating this relationship by parts yields vxdFx vxdGx 0 1 0 1 ∫∫ for all such rx .
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Lecture19-2004 - Lecture XIX Increasing Risk and Ranking...

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