Can MAD (median absolute deviation) or MAE (mean absolute error) be used to calculate prediction intervals?

From my understanding, RMSE (root mean square error) estimated through cross-validation can be used to calculate the prediction interval of a mixed-effect linear model with gaussian error. In my case, the response variable is log10-transformed, so I calculate

10^(RMSE * sigma level)

To estimate the prediction error in terms of orders of magnitude considering a given sigma level (e.g. 1.96 for 95% interval). Can you please confirm this is correct?

Now, I would like to know if I can apply the same calculation to calculate the prediction interval using MAD (median absolute deviation) or MAE (mean absolute error). If not, is there any way to interpret MAE or MAD given a certain level of confidence (e.g. % of times the error is within a given interval)?

Thanks

• In mixed models, you can calculate two types of prediction intervals, one conditional on the random effects, and one from the marginal model that has the random effects integrated out. It is not clear which one are you looking for. Commented Jun 21, 2019 at 8:56
• I'm referring to the one considering fixed effects only Commented Jun 21, 2019 at 14:06

$$\hat{\sigma} = \sqrt{\frac{\pi}{2}}\hat{\text{MAE}}.$$