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Consider two random variables $X$ and $Y$ with joint distribution $F_{X,Y}$ and strictly positive density function $f_{X,Y}$. Additionally, let $x^*$ be the value of $x$ that solves: $$ \Pr[Y\leq y\mid X=x^*]=1-k $$ for some constant $k \in [0,1]$.

I would like to know the minimum conditions I should impose on $F$ in order to ensure that $$\frac{\partial x^*}{\partial k}\geq 0$$

It seems quite intuitive that $COV[X,Y]>0$ would do the job, but this is not the case. I found in the literature that Affiliation* or Decreasing Inverse Hazard Rate** are sufficient, but I would like to know if there are weaker conditions.

*The pdf $f$ is affiliated if $x \leq x'$ and $y \leq y'$ imply $f(x, y')f(x',y)\leq f(x', y')f(x,y)$

** $Y$ is Inverse Hazard Rate Decreasing in $X$ if $\frac{F(y|x)}{f(y|x)}$ is non-increasing in $x$ for all $y$ (where $f(y|x)$ denotes the pdf of $Y$ conditional on $X=x$).

NOTE: This question had also been posted on the Stats Stackexchange site (link here), but was subsequently deleted as it received no comments or answers

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Nothing wrong with posting this here, but there is a stats site in the stackexchange group which might be a better fit. – Gerry Myerson Jul 17 '13 at 12:46
@GerryMyerson: Thanks. I'll try that. Is it generally OK to post the same question in two forums? – EOO Jul 17 '13 at 13:14
It's generally NOT OK to do that. You could 1) wait to see whether you get a good answer here, or 2) delete the question here, and post to the other site, or 3) have it on both sites BUT with a note on each site linking to the post on the other site (transparency!). 4) you could (flag it and) ask a moderator to migrate it to the other site. – Gerry Myerson Jul 17 '13 at 13:19
Thanks -thought so. Will wait and try one of the options later on. – EOO Jul 17 '13 at 13:22
Still waiting? $ $ – Did Nov 11 '13 at 14:54
up vote 1 down vote accepted

In the mathematical statistics literature (for example here) there is a condition termed Regression Dependence. More specifically, for two real random variables $X$ and $Y$, it is said that $Y$ is positively regression dependent on $X$ if: $$ \Pr[Y\leq y |X=x] \quad \text{is non-increasing in }x $$

(Page 1143, Some Concepts of Dependence, E. L. Lehmann. The Annals of Mathematical Statistics, Vol. 37, No. 5 (Oct., 1966), pp. 1137-1153)

This condition is precisely the desired property, namely that $$ \frac{\partial \Pr[Y\leq y |X=x]}{\partial x}\leq 0 $$

As noted on the reference provided for the question, this condition is weaker than Affiliation (and also weaker than Decreasing Inverse Hazard Rate)

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