In my quest to understand all things Spearman, consider the following problem:


Given random variable $x$ with known variance, $\sigma^2$, and $p \in [-1,1]$, one can construct a random variable $y$ such that the Pearson correlation coefficient of the variables $x$ and $y$ is exactly $p$. (Let $y = p x + \sqrt{(1-p^2)} z,$ where $z$ is a random variable independent of $x$ with variance $\sigma^2$.) 


I am wondering if there is an analogue for the Spearman Rank Correlation Coefficient. I am defining the *population* Spearman Correlation Coefficient of the jointly distributed $(x,y)$ as 
$$E[sign((x\_i - x\_j)(y\_i - y\_j))].$$
(this is the subject of another of my questions here on M.O.)


It seems like I would need to know more than just the variance of $x$ to construct $y$, perhaps the entire CDF.



construct random variable with a fixed level of Spearman Coefficient to another
You don't even need $\sigma^2$ to construct such a variable.


Let $Z$ be $+1$ with probability $q$, and $-1$ with probability $1-q$, and independent of $X$.


Let $Y = e^XZ$. This squashes X to the positive reals preserving order, and then may change the sign.


$y\_1$ and $y\_2$ have the same ordering as $x\_1$ and $x\_2$ when the greater value isn't negated. That is, if $x\_1 \gt x\_2$, then $sign((x\_1 - x\_2)(y\_1-y\_2)) = sign(y\_1)$. If $x\_1\lt x\_2$, then $sign((x\_1 - x\_2)(y\_1-y\_2)) = sign(y\_2)$. So, $E[sign((x\_1 - x\_2)(y\_1-y\_2))] = E[Z].$


Choose $Z$ to have average value $p$ (set $q=\frac{(p+1)}2$), and then $X$ and $Y$ have Spearman Rank Correlation Coefficient $p$.


Actually, there is a little ambiguity (to me) about whether you allow $x\_1 = x\_2$, which I ignored above. If under your definition, the rank correlation of $X$ with itself is $\alpha$, then the rank correlation of $X$ with $Y$ is $\alpha p$, and you can get any value in $[-\alpha,\alpha]$.


