RMSE is scale-dependent; is RMSE%? I've got a graph of RMSE% vs. unit size and it declines nicely.  Is this scale-dependence or does the "%" compensate for that?
$$ \text{RMSE%} = 100\% \cdot \frac{\sqrt{\frac{1}{n}\Sigma_{i=1}^n (y_i - \hat{y}_i)^2}}{\bar{y}} $$
 A: A function $f(\cdot)$ is scale-invariant if it yields the same result for argument $x$ as it does for argument $cx$, where $c$ is some positive constant. Let us see whether supplying $(cy_i,c\hat{y}_i)$ in place of $(y_i,\hat{y}_i)$ for $i=1,\dotsc,n$ will change the value of $\text{RMSE%}$:
$$
\begin{equation}
\begin{aligned}
\text{RMSE%}(cy_i,c\hat{y}_i) 
&= 100\% \cdot \frac{\sqrt{\frac{1}{n}\Sigma_{i=1}^n (cy_i - c\hat{y}_i)^2}}{c\bar{y}} \\
&= 100\% \cdot \frac{\sqrt{c^2\frac{1}{n}\Sigma_{i=1}^n (y_i - \hat{y}_i)^2}}{c\bar{y}} \\
&= 100\% \cdot \frac{c\sqrt{\frac{1}{n}\Sigma_{i=1}^n (y_i - \hat{y}_i)^2}}{c\bar{y}} \\
&= 100\% \cdot \frac{\sqrt{\frac{1}{n}\Sigma_{i=1}^n (y_i - \hat{y}_i)^2}}{\bar{y}} = \text{RMSE%}(y_i,\hat{y}_i)
\end{aligned}
\end{equation}
$$
On the way I used the obvious property that $\bar{y}$ gets multiplied by $c$ when $y_i$ is replaced by $cy_i$: if $\bar{y}=\frac{1}{n}\sum_{i=1}^n y_i$, then when scaled by $c$ you get $\frac{1}{n}\sum_{i=1}^n (cy_i)=c\frac{1}{n}\sum_{i=1}^n y_i=c\bar{y}$. 
You see that $\text{RMSE%}$ yields the same result for $(cy_i,c\hat{y}_i)$ as it does for $(y_i,\hat{y}_i)$; hence, it is scale invariant.
