Can anybody think of an example of X and Y that violates the monotonicity condition for $\rho(X)=\sigma[X]-E[X]$? That is, I want to find a pair of rv's $X$, $Y$ such that $X\leq Y$ (the random variable stochastically dominates the random variable ) and $\rho(Y)>\rho(X)$. Thanks in advance!
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Typo in the last inequality. – Eric Towers Oct 25 '21 at 02:03
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Thanks. It's fixed now! – Celine Oct 25 '21 at 02:08
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Do you want $\rho (Y) \leq \rho (X)$ or $\rho (X) \leq \rho (Y)$? Your inequality has simple counter-examples. – Kavi Rama Murthy Oct 25 '21 at 05:07
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I want a pair that violates the condition, so $\rho(X)<\rho(Y)$ – Celine Oct 25 '21 at 10:59
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Let $X=-1$ and $Y=0$. Then, $X\leq Y$, $\sigma(X)=\sigma(Y)=0$ and $\rho(X)=0-(-1)=1$, while $\rho(Y)=0$. – Idontgetit Oct 25 '21 at 11:32
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But that would make $\rho(X)>\rho(Y)$. I am trying to find the opposite! – Celine Oct 25 '21 at 14:24