ABSTRACT
A statistical prediction is a guess about the value of a random variable Y based on the outcome of other random variables X1, . . . , Xm. Thus, a predictor
1 is a (measurable) function, say p(·), of the random variables X1, . . . , Xm. In order to select an optimal predictor, we need a loss function, say ℓ(·), which maps the prediction error to its cost. In principle, the loss function has to be determined case by case, but we can harmlessly assume that if the prediction error is zero also the loss is zero and that ℓ(·) is non-decreasing in the absolute value of the prediction error. Indeed, it is reasonable to assume that an exact guess of the outcome of Y will induce no losses, while the greater the prediction error, the higher the cost.
