ssd-variance - Second Order Stochastic Dominance Definition...

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Second Order Stochastic Dominance Definition Suppose the random variables X and Y have support on [ l, u ]. Then X second-order stochastically dominates Y if Z a l Pr[ X > t ] dt Z a l Pr[ Y > t ] dt for all a . Results 1. If X first order stochastically dominates Y , then X second-order stochastically dom- inates Y . Proof. If X first order stochastically dominates Y , then Pr[ X > t ] Pr[ Y > t ] for all t . Integrating both sides over t gets the desired result. 2. If X second-order stochastically dominates Y , then E [ X ] E [ Y ]. Proof. Recall that E [ X ] = l + Z u l [1 - F ( t )] dt By second order stochatic dominance, this is greater than E [ Y ] = l + Z u l [1 - G ( t )] dt 3. X second-order stochastically dominates Y if and only if E [ h ( X )] E [ h ( Y )] for all increasing and concave function h . Proof. Let h ( t ) = t if t a a if t > a 1
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Obviously h is increasing and concave. Now, E [ h ( X )] = Z a l tf ( t ) dt + Z u a af ( t ) dt = [ - t (1 - F ( t ))] t = a t = l + Z a l [1 - F ( t )] dt + a (1 - F ( a )) = Z a l [1 - F ( t )] dt + l Similarly, E [ h ( Y )] = Z a l [1 - G ( t )] dt + l Hence if E [ h ( X )] E [ h ( Y )], then X
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