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When Monotonic Transformations Preserve Second-Order Stochastic Dominance

Article Quant Q&A · Author: ju k
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Summary

The document poses a question about how transforming two random variables affects second-order stochastic dominance (SOSD). Given that one variable SOSD-dominates another, it asks when applying the same monotonic function to both variables preserves that ordering, and whether sufficient and necessary conditions on the function can be stated.

No answer, proof, example, or empirical evidence is included. The question notes that the property is preserved under linear transformations, but does not specify the relevant restrictions on the linear map or establish what holds for nonlinear functions. The topic is useful for researchers working with distributional comparisons, risk preferences, or transformed financial outcomes, but this document alone does not resolve the conditions. Readers would need a separate reference or derivation to determine which transformations preserve SOSD.

Key ideas

  • The question concerns preservation of second-order stochastic dominance under a shared transformation.
  • It asks whether monotonicity alone is enough to preserve the ordering.
  • The document identifies linear transformations as a case of interest but gives no conditions or proof.
  • No answer or evidence is supplied, so the preservation criteria remain unresolved here.

Tags

StatisticsRisk management
Full text
# Transformation of random variables and second-order stochastic dominance


# Transformation of random variables and second-order stochastic dominance












Suppose $X$ and $Y$ are two random variables where $X$ SOSD* $Y$. Let $g(\bullet)$ be a monotonic function and $X'=g(X)$ and $Y'=g(Y)$. Under what conditions of $g$ is

$X'$ SOSD $Y'$?

I know if $g$ is linear, SOSD property is reserved. Is there any suff and nec conditions of $g$ that assures SOSD property ?

*second-order stochastic dominance.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.