If I want to calculate the RMSE between an estimated value $\hat{x}$ and its reference value $x_{\textrm{ref}}$, let
\begin{equation} y_i = \hat{x}_i-x_{i,\textrm{ref}} \end{equation}
Since
\begin{equation} \textrm{RMSE} = \sqrt{\mathop{\operatorname{mean}}_i(y^2_i)} \quad , \end{equation}
is the standard error of RMSE \begin{equation} \left(\frac{\mathop{\operatorname{stdev}}_i(y^2_i)}{\sqrt{n}}\right)^{1/2} \quad ? \end{equation}
And confidence interval of RMSE
\begin{equation} \left[\ \left(\textrm{RMSE}^2 - t \frac{\mathop{\operatorname{stdev}}_i(y^2_i)}{\sqrt{n}}\right)^{1/2}, \left(\textrm{RMSE}^2 + t \frac{\mathop{\operatorname{stdev}}_i(y^2_i)}{\sqrt{n}}\right)^{1/2}\ \right] \end{equation}
It seems straightforward but I have not found the standard error of RMSE reported in the literature very often. Is it not a meaningful quantity?
If $\sqrt{\operatorname{mean}(y_i^2)}$ is the RMSE, then $\operatorname{stdev}(y_i^2)/\sqrt{n}$ is the (biased estimate of) standard error for RMSE$^2$. However this does not mean you can just take square-roots of everything to get standard error for RMSE, because square-root is a non-linear function.