University of Washington - Department of Statistics
Suppose that we are to predict a future quantity of interest. Our favorite statistical technique provides a predictive probability distribution, say F. However, as it turns out, we are required to issue a single-valued point forecast. How are we going to proceed? By first decision-theoretic principles, our strategy depends on the loss structure. If the loss function is quadratic, the mean of F is the unique optimal point predictor. Under linear loss, we pick the median of F. Under zero-one loss, we pick its mode.
Are standard loss functions realistic? Typically, no. Are standard predictors, such as quantiles, compatible with realistic loss structures? Perhaps surprisingly, typically, yes. Indeed, quantiles arise as optimal point predictors under a general class of economically relevant loss functions, to which we refer as generalized piecewise linear (GPL). The level of the quantile depends on a generic asymmetry parameter that reflects the possibly distinct costs of under-prediction and over-prediction. A loss function for which quantiles are optimal point predictors is necessarily GPL, similarly to the classical fact that a loss function for which the mean is optimal is necessarily a Bregman function.
The applied relevance of these results will be illustrated using the Bank of England's density forecasts of United Kingdom inflation rates, and probabilistic predictions of wind speed at a Pacific Northwest wind energy site.