Role of expected loss of the best forecast in decision theory

Question

Suppose we have a random variable $Y$ with an unknown distribution $P$. We model it with a distribution $Q$. We are asked to make a point forecast under some type of loss $L$. We choose the loss minimizing $\hat y=\arg\min_y\int L(s-y) dQ(s)$ as our forecast. I am interested in the quantity

• $\int L(s-\hat y) dP(s)$ which is the expected loss of this point forecast w.r.t. the true distribution of $Y$

and to a lesser extent in

• $\int L(s-\hat y) dQ(s)$ which is the expected loss of this point forecast w.r.t. our model.

Question: Does any of these quantities have a name and do they play a role in any of the major decision theoretic frameworks?

I would like to learn more about their roles and intepretations; connections to economic literature are welcome.

(More generally, I am interested in decision making under uncertainty when $P$ and $Q$ do not coincide and am looking for theory/theories that account(s) for this in detail. References are welcome.)

Answer 1

A good start would be a couple of papers by Tilmann Gneiting, e.g. Gneiting (2011), "Quantiles as Optimal Point Forecasts", International Journal of Forecasting, or also Gneiting (2011), "Making and Evaluating Point Forecasts", JASA. He is mainly writing about $Q$ and about minimizing the expected loss for various loss functions $L$ (although not so much about the loss itself). I have an accepted commentary on the M4 forecasting competition in the IJF where I make similar points to argue that evaluating the same point prediction with different loss functions ("error measures" in forecasting terminology) - which is very common in forecasting - makes little sense.

The pretty much accepted nomenclature among forecasters for the loss is "loss", and for the optimal point forecast is "optimal point forecast". (Sorry.) I am unfortunately not aware of any work specifically on the expected loss under the true density $P$.

I would be interested in any literature on the questions you raise, especially anything that relates the loss under the true $P$ to $Q$ or some measure of distance between the two (Kullback-Leibler looks like it might figure here).