Some Thoughts About the Design of Loss Functions

The choice and design of loss functions is discussed. Particularly when computational methods like cross-validation are applied, there is no need to stick to “standard” loss functions such as the L2-loss (squared loss). Our main message is that the choice of a loss function in a practical situation is the translation of an informal aim or interest that a researcher may have into the formal language of mathematics. The choice of a loss function cannot be formalized as a solution of a mathematical decision problem in itself. An illustrative case study about the location of branches of a chain of restaurants is given. Statistical aspects of loss functions are treated, such as the distinction between applications of loss functions to prediction and estimation problems and the direct definition of estimators to minimize loss functions. The impact of subjective decisions to the design of loss functions is also emphasized and discussed.