The ETS function in the fable package for R provides an argument called opt_crit that specifies the quantity that is minimized during parameter estimation (there is also similar functionality in the forecast package). One of the options for this is "amse", which specifies "Average MSE over first nmse forecast horizons". Is this a standard procedure for estimating time series forecasting models? Are there references evaluating this approach and comparing to maximum likelihood estimation? I didn't see anything about this in "Forecasting with Exponential Smoothing: The State Space Approach" by Hyndman et al. or other sources I have available.
The AMSE as an estimation criterion was introduced in Hyndman, Koehler, Snyder & Grose (IJF, 2002). The code was originally written for that paper, which is why the AMSE option is there. I haven't used it any subsequent papers, and as you found, we didn't include it in the 2008 Springer book.
It is frequently used as an evaluation criterion, but under the name MSE.