# Unique matching for quantiles from one half of the density to a subset of the other half?

I am interested in finding the median absolute distance to quantiles. So, for $Q_\alpha$ the $0 \le \alpha \le 1$ quantile, I would like to find $Q_\gamma^*$ such that $Q_\gamma^*$ satisfies \begin{equation} \underset{\gamma}{median}|Q_\alpha-Q_\gamma|= |Q_\alpha-Q_\gamma^*|. \end{equation}

Consider a symmetrical distribution, say the Gaussian. Then for the median quantile $Q_{0.5}$ there are two possible $Q_\gamma^*$ at equal absolute distance from it. For our purposes, let's only consider the one where $\gamma >0.5$. This is an upper bound in the following sense: For all quantiles less than the median, their $Q_\gamma^*$ is less than (or equal to?) the $Q_\gamma^*$ of the median. In fact, for $Q_0$, $Q_\gamma^* = Q_{0.5}$.

Here is my question. To the left of the median is 50% of the density of the distribution, but it seems like the range of possible $\gamma$ values giving $Q_\gamma^*$'s for $0 \le \alpha \le 0.5$ spans only a portion of the density to the right of the median (from $Q_{0.5}$ to $Q_{\gamma,0.5}^*$). So it seems as though there must be ties in the $Q_\gamma^*$ for some of the unique $Q_\alpha$'s. On the other hand, if the distribution is has finite support, can you always find a unique $Q_\gamma^*$ for a given $Q_\alpha$?

• I don't think I understand the argument about ties. You seem to argue that because the function $\alpha\to\gamma^{*}$ from $[0,1/2]$ to $[1/2, 1]$ is not surjective, then it cannot be injective. But a consideration of such functions in general, such as $\gamma^{*}(\alpha) = 1/2 + \alpha/2$, shows that this conclusion is not necessary. What exactly do you mean by "infinitely dense"? – whuber Oct 21 '14 at 7:05
• I think there might be ties because quantiles to the left of the median constitute 50% of the density. The $Q_{\gamma,0.5}^*$ is the largest $Q_{\gamma}^*$ to the left of the median, and $Q_{0.5}$ is the smallest. It doesn't seem like there are enough possible $Q_\gamma^*$ on the right side of the median to give unique matches for all the $Q_\alpha$ on the left side. – Deathkill14 Oct 21 '14 at 7:46
• One thing that encourages me to believe a unique mapping can be found is that I can for example uniquely map $(-\infty, \infty)$ to (0,1). – Deathkill14 Oct 21 '14 at 8:14
• Are you talking about quantiles of a distribution, as written in the question, or about quantiles of a finite sample (which is the only way in which I can make sense of your comments)? – whuber Oct 21 '14 at 17:32
• I have the quantiles of a distribution in mind. Sorry. This is all at the distributional level. – Deathkill14 Oct 22 '14 at 19:43