TY - JOUR
T1 - Unrestricted and controlled identification of loss functions
T2 - Possibility and impossibility results
AU - Lieli, Robert P.
AU - Stinchcombe, Maxwell B.
AU - Grolmusz, Viola M.
N1 - Publisher Copyright:
© 2019 International Institute of Forecasters
PY - 2019/7/1
Y1 - 2019/7/1
N2 - The property that the conditional mean is the unrestricted optimal forecast characterizes the Bregman class of loss functions, while the property that the α-quantile is the unrestricted optimal forecast characterizes the generalized α-piecewise linear (α-GPL) class. However, in settings where the forecaster's choice of forecasts is limited to the support of the predictive distribution, different Bregman losses lead to different forecasts. This is not true for the α-GPL class: the failure of identification is more fundamental. Motivated by these examples, we state simple conditions that can be used to ascertain whether loss functions that are consistent for the same statistical functional become identifiable when off-support forecasts are disallowed. We also study the identifying power of unrestricted forecasts within the class of smooth, convex loss functions. For any such loss ℓ the set of losses that are consistent for the same statistical functional as ℓ is a tiny subset of this class in a precise mathematical sense. Finally, we illustrate the identification problem that is posed by the non-uniqueness of consistent losses for the moment-based loss function estimation methods proposed in the literature.
AB - The property that the conditional mean is the unrestricted optimal forecast characterizes the Bregman class of loss functions, while the property that the α-quantile is the unrestricted optimal forecast characterizes the generalized α-piecewise linear (α-GPL) class. However, in settings where the forecaster's choice of forecasts is limited to the support of the predictive distribution, different Bregman losses lead to different forecasts. This is not true for the α-GPL class: the failure of identification is more fundamental. Motivated by these examples, we state simple conditions that can be used to ascertain whether loss functions that are consistent for the same statistical functional become identifiable when off-support forecasts are disallowed. We also study the identifying power of unrestricted forecasts within the class of smooth, convex loss functions. For any such loss ℓ the set of losses that are consistent for the same statistical functional as ℓ is a tiny subset of this class in a precise mathematical sense. Finally, we illustrate the identification problem that is posed by the non-uniqueness of consistent losses for the moment-based loss function estimation methods proposed in the literature.
KW - Bregman loss functions
KW - GPL loss functions
KW - Identification
KW - Loss functions
KW - Osband's principle
KW - Point forecasts
UR - http://www.scopus.com/inward/record.url?scp=85063091271&partnerID=8YFLogxK
U2 - 10.1016/j.ijforecast.2018.11.007
DO - 10.1016/j.ijforecast.2018.11.007
M3 - Article
AN - SCOPUS:85063091271
SN - 0169-2070
VL - 35
SP - 878
EP - 890
JO - International Journal of Forecasting
JF - International Journal of Forecasting
IS - 3
ER -