I believe that the values I am forecasting are gamma distributed with shape $k>0$ and scale $\theta>0$. I need a point forecast (i.e., a one-number summary) that minimizes the expected error. What point forecast does so, if my error measure is
- the (mean) squared error (MSE)?
- the (mean) absolute error (MAE)?
- the (mean) absolute scaled error (MASE)?
- the (mean) absolute percentage error (MAPE)?
I am asking and self-answering to have a reference for the future, since I often use this as an example to illustrate the properties of different error measures, e.g., at What are the shortcomings of the Mean Absolute Percentage Error (MAPE)? The analogous question for the lognormal distribution can be found here.