ABSTRACT
A ‘piecewise linear’ expected loss can be matched quite closely by a ‘piecewise constant’ expected loss, although the corresponding parameters usually differ substantially. Similarly, to a linear loss, a piecewise quadratic loss function is defined by its elements, or segments, each of them a quadratic function given by the threshold and a coefficient. Further flexibility is introduced by convex combinations of loss functions. Of particular importance are combinations with piecewise constant loss functions. The expected loss with the piecewise linear loss is evaluated using the by-now familiar calculus. The piecewise quadratic loss is motivated similarly by reflecting the uneven consequences of the positive and negative errors. An alternative for the first property is the mean absolute error, in which the loss is equal (or proportional) to the absolute value of the error. The quadratic loss has some affinity with its namesake in established forms of estimation, as applied in the mean squared error.
