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Stochastic dominance is a partial order between random variables. [1] [2] It is a form of stochastic ordering.The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers.
Downward-lexicographic dominance, denoted , means that has a larger probability than of returning the best outcome, or both and have the same probability to return the best outcome but has a larger probability than of returning the second-best best outcome, etc. Upward-lexicographic dominance is defined analogously based on the probability to ...
Ranking gambles by mean-preserving spreads is a special case of ranking gambles by second-order stochastic dominance – namely, the special case of equal means: If B is a mean-preserving spread of A, then A is second-order stochastically dominant over B; and the converse holds if A and B have equal means.
the family exhibits the first-order (and hence second-order) stochastic dominance in , and the best Bayesian update of is increasing in (). But not conversely: neither monotone hazard rates nor stochastic dominance imply the MLRP.
Monotone with respect to first order stochastic dominance. If is a concave distortion function, then is monotone with respect to second order stochastic dominance. is a concave distortion function if and only if is a coherent risk measure. [1] [2]
The presence of marginal conditional stochastic dominance is sufficient, but not necessary, for a portfolio to be inefficient. This is because marginal conditional stochastic dominance only considers incremental portfolio changes involving two sub-groups of assets—one whose holdings are decreased and one whose holdings are increased.
The test does not identify where this stochastic dominance occurs or for how many pairs of groups stochastic dominance obtains. For analyzing the specific sample pairs for stochastic dominance, Dunn's test, [4] pairwise Mann–Whitney tests with Bonferroni correction, [5] or the more powerful but less well known Conover–Iman test [5] are ...
The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function (+, +) to lie in the second-order cone in +. [ 1 ] SOCPs can be solved by interior point methods [ 2 ] and in general, can be solved more efficiently than semidefinite programming (SDP) problems. [ 3 ]