<p>As long as <span class="math-container">$g(\cdot)$</span> is <strong><em>Nondecreasing</em></strong> and <strong><em>Concave</em></strong></p>

<p>Definition: For any lotteries <span class="math-container">$F$</span> and <span class="math-container">$G$</span>, <span class="math-container">$F$</span> second-order stochastically dominates <span class="math-container">$G$</span> if and only if the decision maker weakly prefers <span class="math-container">$F$</span> to <span class="math-container">$G$</span> under every weakly <em>increasing concave</em> utility function <span class="math-container">$u$</span>. <a href="https://ocw.mit.edu/courses/economics/14-123-microeconomic-theory-iii-spring-2015/lecture-notes-and-slides/MIT14_123S15_Chap4.pdf" rel="nofollow noreferrer">Link</a></p>

CC BY-SA 4.0
2019-04-20T17:40:29.110Z
2019-04-20T17:40:29.110Z
S. Phil Kim
<p>Suppose <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are two random variables where <span class="math-container">$X$</span> has SOSD (second order stochastic dominance) over <span class="math-container">$Y$</span>. Let <span class="math-container">$g(\cdot)$</span> be a monotonic function and <span class="math-container">$X'  = g(X)$</span> and <span class="math-container">$Y' = g(Y)$</span>. </p>

<p>Under what conditions of <span class="math-container">$g(\cdot)$</span> <span class="math-container">$X'$</span> has SOSD over <span class="math-container">$Y'$</span>? I know that if <span class="math-container">$g$</span> is linear, SOSD property is reserved. Is there any sufficient and necessary conditions of <span class="math-container">$g$</span> that assures SOSD property?</p>

<statistics><decision-theory><risk>
Transformation of random variables and second order stochastic dominance
user20775
# Transformation of random variables and second order stochastic dominance

(Asked by: username_0 on 2019-01-15)

Suppose $X$ and $Y$ are two random variables where $X$ has SOSD (second order stochastic dominance) over $Y$. Let $g(\cdot)$ be a monotonic function and $X' = g(X)$ and $Y' = g(Y)$.

Under what conditions of $g(\cdot)$ $X'$ has SOSD over $Y'$? I know that if $g$ is linear, SOSD property is reserved. Is there any sufficient and necessary conditions of $g$ that assures SOSD property?

**Tags:** statistics, decision-theory, risk

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## 1 answers

### Answer by username_1 on 2019-04-20. Score: 1

As long as $g(\cdot)$ is ***Nondecreasing*** and ***Concave***

Definition: For any lotteries $F$ and $G$, $F$ second-order stochastically dominates $G$ if and only if the decision maker weakly prefers $F$ to $G$ under every weakly *increasing concave* utility function $u$. [Link](https://ocw.mit.edu/courses/economics/14-123-microeconomic-theory-iii-spring-2015/lecture-notes-and-slides/MIT14_123S15_Chap4.pdf)


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